Miniaturized Stepped-Impedance Resonator Band Pass Filter Using Folded SIR for Suppression of Harmonics

In the recent trends in communication, such as green communication and radar systems, removing RF noises becomes an essential quality to be taken into the care of for better performance [1]. The planar bandpass filters [2] are the major components in the current 5G wireless communication, which utilize spectrum bands such as 700 MHz, 3.57 GHz, and 25.8 GHz [3]. Various techniques such as couple line, hairpin, combine, step, and stub impedance [4] are reported in the literature.

Many have talked about making printed planar microwave filters smaller in the last few years. This is because low-frequency wireless and mobile communication systems are becoming increasingly common. Examples of this type of research can be found in references [5-7]. Because they work at low frequencies, conventional distributed resonators are too big to work well. As a result, larger structures have been phased out in favor of smaller ones. Hairpin resonators [8] were replaced for half the wavelength resonators, which were used to create parallel-coupled line resonating filters [9]. Open-loop square resonators [10] are a more compact alternative to the hairpin resonator, though this resonator still functions as a distributed device and produces excessively complex layouts at lower frequencies. The presence of unwanted spurious harmonics reduces the amount of out-of-band rejection, another disadvantage of distributed resonators. Compact quasi-lumped resonators can be used to overcome this issue, but the increased losses they incur are a trade-off. There is a lot more capacitance in which the hairpin resonator is folded [11] than in conventional hairpin or open-loop resonators, which makes it smaller. It is possible to obtain similar findings by employing the open-loop impedance resonators suggested in references [12, 13], now known as folded Stepped-Impedance Resonators (SIRs) [14]. It's possible to think of this resonator as a line with a low-impedance pair of lines linked with a high characteristic impedance. As an LC quasi-lumped resonator, the structure works at low resonance frequencies for capacitance values in the section of the wide coupled line that is relatively high. Inductance (L) comprises short sections of high-characteristic-impedance microstrip lines, and a short pair of electrically connected microstrip lines make up the rest and provides the capacitance (C). The resonator's quasi-lumped design automatically eliminates the problem of a first spurious transmission band. More advanced SIRs have been applied to filter design to achieve compactness and an expansive upper stopband.

Short snippets of a quarter wavelength When the vertical size of the BPF needs to be decreased, it is common practice to employ a BPF rather than a quarter-wavelength open stub BPF [24]. However, in BPFs with λg/2 open stubs or λg/4 short stub (18.7 mm×18.9 mm at 5.8 GHz and 0.45 λg×0.19 λg), there is no design that can minimize the transmission line’s (TL) horizontal-size [25]. Currently, a dielectric ($\varepsilon_r$) substrate of high value is utilized to shrink the size of a BPF, but this method is prohibitively expensive. Therefore, there is a pressing need to develop BPFs with desirable characteristics such as high response, low cost and compact size [26].

Numerous studies have been conducted to minimize the footprint of stub BPFs while yet providing abundant bandwidth [27]. Zeroth-order resonance (ZOR) was employed by Palanisamy et al. [28] to create a BPF measuring 12.9 mm×5.9 mm with a bandwidth of 120%. However, coupling losses were produced since this BPF employed feeding line of coupling structure. The proposed BPF in reference [29] had an in/output coupling structure and used the defective ground structure (DGS) method to achieve a bandwidth of 116%. This BPF had a number of drawbacks, including its large size, the coupling losses it caused, and the parasitic characteristic it caused between the ground and transmission line. In reference [30], a BPF with a 109% bandwidth and a dimension of 23.4 mm 10 mm was presented. The DGS method was also utilized by this BPF. However, it had poor insertion losses and poor parasitic performance between the transmission line and ground. Broadband pass filters (BPFs) of large dimensions (123% [31] and 146%) were disclosed in the study [32]. These BPFs ranged in size from 32 mm 10 mm to 13.0 mm 10.0 mm to 150 mm 150 mm. High insertion losses (0.8 dB [33], 0.3 dB [34], and 0.55 dB [35]) were another drawback of these devices.

Using a folded SIR structure and T-shaped SIR transmission line, we present a small BPF with λg/4 short circuited stubs. Transmission lines in the shape of a T and folded SIR are planned to cut down on both the horizontal and vertical lengths. Folded and T-shaped SIRs replace the protruding series λg/4 transmission line and shunt λg/4 short stubs along the vertical and horizontal axes, respectively, to reduce the overall footprint of the proposed BPF. The small a BPF is easy to build and has negligible insertion losses.

The guided wavelength and the effective dielectric constant values are essential in designing slot in a dielectric substrate. Consider λ_cutoff is the cut off wavelength, the guided wavelength (λ_guided) of the slot is given by:

$\lambda_{\text {guided }}=1 / \sqrt{\left(\frac{1}{\lambda_0}\right)^2-\left(\frac{1}{\lambda_{\text {cutoff }}}\right)^2}$    (1)

The slot antenna's effective dielectric constant (ε_eff) is calculated as follows:

$\varepsilon_{\text {eff }}=\frac{\varepsilon_{\text {reff }}+1}{2}$    (2)

Consider the angular cut off frequency as ω_c, angular resonant frequency as ω₀, g₁ is the prototype value of the Butterworth small pass filter type, Z₀ is the characteristic impedance, c is the swiftness of light then the inductance (L) and capacitance (C) values shall be found by:

$L=\frac{1}{4 \pi^2 c f_0^2}$    (3)

$C=\frac{\omega_c}{Z_0 g_1} \cdot \frac{1}{\omega_0^2-\omega_c^2}$   (4)

The bandwidth (B) of the wideband patch antenna is the difference between the lower (f_Lower) and higher (f_Higher) cutoff frequencies:

$B=f_{\text {Lower }}+f_{\text {Higher }}$    (5)

The central frequency (F_Central) along the antenna bandwidth is given by:

$F_{\text {Central }}=\left(f_{\text {Lower }}+f_{\text {Higher }}\right) / 2=\sqrt{f_{\text {Lower }} f_{\text {Higher }}}$    (6)

Fractional bandwidth (BW) is given by the ratio of the general bandwidth of the antenna to its central frequency:

BW=B/F_Central    (7)

Fractional bandwidth (BW) shall be expressed in terms of percentage as:

% BW=(B/F_Central)×100%    (8)

The different ground plane structures with several length sizes like full length L, three quarters length 3L/4, half-length L/2 and quarter length L/4 are shown in Figure 1. The ground plane is reduced to three fourth of its size and then to half for widening the bandwidth. The return loss plot of the variety of defected ground structures used in this antenna design.

1.png

Figure 1. Circuit scheme of tunable transmission zeroes by tapping the input and output resonators

2.png

Figure 2. With floating conductor and ground plane window representing folded stepped impedance resonator

Because of the lines that run parallel to each resonator, the general layout of each microstrip circuit is shown in Figure 1. Because of the adjustment, the value of this device's input and output impedances has increased by a factor of three due to the change. Both the input and output ports use step-impedance microstrip lines, which are placed close to the characteristic impedance Z₀. Figure 2 illustrates the standard SIR structure. On each resonator's resonator, voltage distributions can be either asymmetric (referred to as odd mode) or symmetric (referred to as even mode). It is possible to estimate the resonance frequencies of the SIR, as stated in reference [29].

$\tan \theta_1=\operatorname{Rcot} \theta_2\left(\right.$ odd mode at $\left.f=f_1\right)$    (9)

$\cot \theta_1=-R \cot \theta_2\left(\right.$ odd mode at $\left.f=f_2\right)$   (10)

where, SIR impedance ratio, $R=\frac{Z_2}{Z_1}$.

Higher-order modes: T-shaped SIRs can support higher-order modes at multiples of the fundamental frequency, potentially contributing to unwanted harmonics in the filter response.

Harmonic suppression analysis: Investigate how design parameters (impedance steps, J-inverter admittance, coupling) influence the excitation and attenuation of higher-order modes. Techniques like modal analysis or studying the frequency dependence of the characteristic equation and S-parameters may be employed.

Stopband rejection: Assess stopband rejection at harmonic frequencies, with a specific focus on achieving the desired suppression level and understanding its dependence on design choices.

Non-idealities: Incorporate considerations for non-idealities such as conductor losses, radiation, and substrate effects in the analysis for more precise predictions.

Optimization: Apply optimization techniques to identify design parameters that achieve the desired performance at both fundamental and harmonic frequencies.

The ground plane window provides these additional benefits. Thus, we can now design filters with increased bandwidths and improved rejection levels above the first passband that are smaller than those based on conventional folded SIR resonators. Folded SIR with a floating conductor is an improved version of the proposed resonators. There is a metallic floating rectangular patch below the coupled lines section within the ground window due to parting with part of the ground plane below the resonator. Because of the window, the single strip section has a higher characteristic impedance, which reduces its size [33]. A more critical consideration in this paper concerns the floating conductor's ability to enhance the Filters' undesired band rejection level depending on this floating conductor structure [34]. The prevention of intermodulation distortion necessitates attention to this issue. First, the desired pass band's initial resonance frequency is close to the first resonance frequency of the typical folded SIR. At the same time, the 2^nd resonating frequency band is meaningfully raised. Because of this, the first band can be better rejected. By opening a novel coupling mechanism, which improves the cohesion between nearby resonators [27], the ground plane window provides additional advantages, such as increased bandwidth. Thus, we can now design filters with increased bandwidths and improved rejection levels above the Ist pass band that are smaller than those based on conventional folded band pass SIR resonators.

5.1 Single resonator characterization

In this study, the model for the folded SIR with a floating conductor is simply an approximation because the coupled section has three conductors plus ground. There are now three quasi-TEM modes rather than the previously stated two. According to this theory, however, this is a practical application [28, 31]. A single transmission line section that contributes inductively has a characteristic impedance and ε_eff, an effective dielectric constant. Ze and Zo, as well as the effective modal impedances of the even (e) and odd (o) mode and effective permittivity $\varepsilon_{\text {eff }}^e \varepsilon_{\text {eff }}^0$, are the electrical parameters of the capacitive part of the resonator. This is the identical folded hairpin resonator circuit proposed by Lee et al. [32]. Afterward, two distinct resonance situations can be found:

$\tan \frac{\theta_s}{2} \tan \theta_0=R z_o$ (odd resonances)

$\tan \theta_e=-R z_e \tan \frac{\theta_s}{2}$ (even resonances)          (12)

To calculate Rz_o= $\frac{Z_O}{Z_S}$, Rz_e= $\frac{Z_e}{Z_s}$, where $\theta_s$ is the single line section's electrical length, θ_ethe coupling line section's even (odd) mode's electrical length. The first odd mode resonance, f₀, occurs at this frequency. Using these resonators as a basis, we can determine the frequency range of any filter that employs them. It is f₁ relates to the first even mode resonance, the first spurious resonance.

5.2 Bandpass filter design

Take, for example, the following specifications for filter A: There are three orders of magnitude (N), a single-core frequency (f₀=4 GHz), ripple (0.1 decibels), and a fractional bandwidth of 4%. The same substrate was used as in the previous part. Each filter was contained within a copper box with dimensions of 40×20 mm². Thickness of 2.0mm was chosen for the air layers beneath and over the substrates, respectively. However, it is critical to emphasize that the metallic enclosure has a negligible effect on the first spurious band and pass band due to the severely limited electromagnetic environment surrounding metallic strips and slots. The printed metallization controls the filter's response at particular frequencies, but the enclosure has little effect on the filter's response. Bear in mind that this is not always true in the case of other deferred substrate structures [33, 34], where boundary dimensions are essential to design. This is a practical novelty and uniqueness of our design, as it enables the filter to be integrated as a component of a more intricate system without requiring changes to the system's physical structure.

It will be shown later that purposefully made compact housing can reduce microstrip line levels between the intended and spurious bands. You should keep the housing size manageable if you don't want spurious resonances above and near the pass band and lower the rejection level. When it comes to radiation, it is only relevant in the spurious band frequency zone. In the trials, the folded SIRs were folded to have the following parameters with dimensions: L₂=4.1 millimeters. w₀=0.46 millimeters, l₁=5.27 millimeters, values of l₃ that resonates at f₀ for each resonating filter are as follows: l₃=5.8 mm (conventional folded Stepped Impedance Resonator) and l₄=3.8 mm (convolutional folded Stepped Impedance Resonator) (folded Stepped Impedance Resonator with floating conductor). When using a floating conductor, S=0.25 mm, w_f=2.7 mm, and l=6.85 mm is all valid values.

As a point of comparison, we've included a design in which the width of the strip is also 0.25 mm, and the length of the microstrip is the same. The required to suit the resonance frequency is 64.4 mm. Following the typical design process based on estimating the coupling coefficient between nearby resonators, we have implemented our system. The coupling factor represents that distance between the resonators is the relaying function, denoted by the letter d. Unless the separations are minimal, it is possible to see an intriguing extra feature of the resonator presented in this paper: Traditional SIR folded in half, or open-loop impedance resonator has a lower coupling factor. Thus, the proposed geometry is less sensitive to dimensional constraints for a given bandwidth, allowing for greater bandwidths in the opposite direction. The specifications of filter A result in a coupling factor of K=0.037 due to the design. This necessitates the use of traditional folded SIRs that are incredibly close together. There are floating rectangular patches in the space between the resonators. On the other hand, fabricating is rather large and reasonably simple. For the in-out resonators, the external quality factor is used. Figure 3 illustrates the simulated broadband insertion losses of filters A, B, and C constructed using the resonators above.

3.png

Figure 3. S parameter of the traditional SIR filter

Consider designing a filter around the suggested resonator with greater relative bandwidth than the original resonator. The following specifications characterize filter B: central frequency f₀=4.2GHz, order N=4, ripple=0.10 decibels, and fractional bandwidth=7%. Because the substrate is similar to the prior designs, no changes are necessary. The resonator geometry parameters are as follows: l₂=2.08 mm, l₁=3.62 mm, l₃=4.01 mm, i_f=3.03 mm, l₃=4.35 mm, w_f=2.4 mm and s=0.35 mm. The filter specs include the coupling factors K₁₂=0.045 and K₂₃=0.039, respectively.

As a consequence of these values, the gap sizes are determined to be d₁₂=0.3 mm and d₂₃=0.34 mm, respectively. We can create the same filter using the traditional folded SIR with a gap between the resonators less than 30 micrometers wide. Consequently, even with more exact methods of fabrication, tolerances in dimensions would continue to play an increasingly significant role. A decent figure for the in-out resonators is Q=21.3, the external quality factor. The distance that must be maintained between the input and output resonators, which is equal to 3.71 millimeters, serves as the criterion for determining where the 45-degree feed lines, each of which has a width of 0.35 millimeters, must be positioned over the sides of both the input and output resonators to arrive at this value. Figure 4 presents a representation of the passband response provided by the filter. At a frequency of about 5.3 GHz, the insertion loss curve hits its highest point of 30 dB of maximum loss.

4.png

Figure 4. S parameter of the folder SIR filter

5.png

Figure 5. S parameter of the full wave simulated folded SIR BPF filter

Box size can be arbitrarily adjusted to achieve high rejection levels above and beyond the frequency range of a first-pass band. It has been possible to do a full-wave simulation of filter B using HFSS for two different box sizes. For comparison, an ensemble simulation for an endlessly broad box is presented. It should be noted that, for a box width of 19 mm, the Ensemble and HFSS values are highly similar to one another deficient transmission level is possible between the spurious passbands and the desired band when using a box width of 9 mm. The enclosure supports quasi-TE10 mode and is placed distant from the enclosure's cutoff point, in this case, of 9 mm width. In this mode, transmission is impossible because of the extraordinarily high reactive attenuation of the fields involved. In addition, about 80 dB of attenuation is achieved. The output port's evanescent electromagnetic field is strong enough to limit attenuation to 40-50 dB, even for a 21 mm wide box. Figure 5 shows insertion loss response of the full wave folded SIR Filter. A small effect of the lateral walls on the pass bands may be seen, but the enclosure's waveguide mode features strongly influence the rejection level between pass bands. Final thoughts on the transmission zeros identified in measurements and simulations but not considered by the filter's equivalent circuit. As a result of destructive interference between the microstrip system and other weak transmission strategies mechanisms, such as transient fields in the box or surface waves, these zeros are generated. The circuit model does not represent either of these methods at this time. Only at frequencies in the circuit model predicts minimum transmission levels, which is not very frequent, is this interference meaningful.

The proposed BPF is compared with others in Table 2, where parameters such as bandwidth, insertion loss, and overall device size are evaluated.

Table 2. Comparative analysis of S-parameters, size, bandwidth with proposed filter