In this paper, the weighting techniques are directly acted on traditional 1D sinusoidal electromagnetic bandgap (EBG) structures. The proposed 1D weighted EBG band-stop filter is theoretically designed by using coupled-mode theory, and the coupling coefficient amplitudes of the 1D sinusoidal EBG structure are directly weighted using window functions. The passband ripple, reflected sidelobes, and bandwidth of the band-stop filter are improved using weighted window functions, and the bandgap depth is adjusted by changing the amplitude factor of weighting functions. For validation, the fabrication and measurements have also been done for the proposed filter. The measurement and simulation results have a close match, and the design concept is verified.

Band-stop filters play important roles in communication systems. Several band-stop filters have been developed recently.1–5 Electromagnetic bandgap (EBG) structures are widely used in microwave systems. The two-dimensional (2D) EBG filters have been fabricated by etching rows of particular unit cells, such as circular6 or square7 periodic holes in the ground plane. 1D EBG structures based on the sinusoidal impedance modification of the shape of the microstrip line and without etching in the backside metallic ground plane were realized.8–10 The analytical solution based on coupled-mode theory is applied to the design of the 1D sinusoidal EBG structure.11–13 1D sinusoidal EBG structures have the important advantage of single bandgap without higher order harmonics. Consequently, the single-band function is conveniently realized.6 Meanwhile, 1D EBG structures are easily fabricated and integrated with monolithic microwave integrated circuits. However, the reflected sidelobes and passband ripple of the 1D sinusoidal EBG filter cannot be neglected and need to be suppressed.13,14

In this paper, weighting windows have to directly adjust the 1D sinusoidal microstrip structure by using coupled-mode theory. The bandgap depths are changed by changing the amplitude factors of weighting functions. The weighted technologies improved the passband ripple and reflected sidelobes, and the bandwidth of band-stop filters becomes wider.

Using coupled-mode theory15,16 and assuming TEM or quasi-TEM single-mode operation (as it happens in most of the planar circuit technologies), the electromagnetic behavior of an EBG structure can be characterized through the coupling coefficient, K(z), between the forward and backward traveling waves associated with the operation mode. If the EBG structure is periodic along the propagation axis z with period Λ, then K(z) produced by the perturbation is also periodic with the same period, and it can be expanded in a Fourier series. To ease the design process and obtain analytical design expressions, a sinusoidal profile for K(z) is employed. Therefore, all the coefficients are equal to zero except for K±1 = A/2, where A is the amplitude of the sinusoidal coupling coefficient. Thus, the characteristic parameters for the two-port EBG structure band-stop filter, including the central frequency (f0), rejection level [S21min], return level [S11max], and bandwidth (BW) between zeros, can be expressed as follows:16,17

f0=c2Λεeff,
(1)
S21min=sech(AL/2),
(2)
S11max=tanh(AL/2),
(3)
BW=cπεeffA22+πL2,
(4)

where c is the speed of light in vacuum, L is the device length, and ɛeff is the effective dielectric constant.

The characteristic impedance profile, Z0(z), is obtained using the designed two-port EBG structure with the sinusoidal coupling coefficient K(z),

Z0(z)=Z0(0)eAΛπsin2πΛz+θsinθ,
(5)

where Z0(0) is the characteristic impedance at the input port of the device.

The original-unweighted 1D sinusoidal EBG filter [as shown in Fig. 1(a)], which was regarded as a comparison structure, was a EBG filter with the stop-band central frequency f0 = 3.6 GHz, rejection level S21 = 32 dB, and rejection bandwidth BW = 2.4 GHz. The substrate used is a F4B-2 woven-glass-filled, polytetrafluoroethylene (PTFE) copper-clad substrate, with ɛr = 3.5 ± 0.1 and thickness h = 1.5 mm. The initial phase is fixed at θ = 0°, and the characteristic impedance of input and output ports is Z0 = 50 Ω. Then, the length of the filter and the amplitude of the sinusoidal coupling coefficient are determined as follows:

L=5Λ=125mm,
A=70.341m1.
FIG. 1.

Diagram of the original-unweighted sinusoidal EBG structure (a) and simulated S21 parameters (b) of 1D EBG microstrip structures with a single weighted amplitude factor of Rmax = 1.

FIG. 1.

Diagram of the original-unweighted sinusoidal EBG structure (a) and simulated S21 parameters (b) of 1D EBG microstrip structures with a single weighted amplitude factor of Rmax = 1.

Close modal

Different weighting windows are applied to the sinusoidal microstrip structure.18,19 According to Eqs. (2) and (3), the S21 and S11 parameters can be changed by changing the amplitude of the sinusoidal coupling coefficient, A, while the length of the filter is fixed. The weighting functions are directly tuned on the amplitude of the sinusoidal coupling coefficient, and the weighted amplitude of the coupling coefficient of the nth sinusoidal period, An, is shown in Table I.

TABLE I.

Coupling coefficient amplitude with a weighted function.

Weighting function Weighted amplitude of the 
type coupling coefficient of the 
 nth period unit, Anz′ ∈ [−1, 1] 
Blackman An=Rmax(0.42+0.5cos(2πzn) 
 +0.08cos(4πzn))A 
Gaussian An=Rmaxexp(2(2zn)2)A 
Hamming An=Rmax(0.54+0.46cos(2πzn))A 
Hanning An=Rmax(0.5(1+cos(2πzn)))A 
Weighting function Weighted amplitude of the 
type coupling coefficient of the 
 nth period unit, Anz′ ∈ [−1, 1] 
Blackman An=Rmax(0.42+0.5cos(2πzn) 
 +0.08cos(4πzn))A 
Gaussian An=Rmaxexp(2(2zn)2)A 
Hamming An=Rmax(0.54+0.46cos(2πzn))A 
Hanning An=Rmax(0.5(1+cos(2πzn)))A 

Here, Rmax represents the weighted amplitude factor. For Rmax = 1, Fig. 1 depicts the S21 parameters of original-unweighted and weighted EBG filters simulated using 3D-CST-MicrowaveStudioTM. It is obvious that the S21 scattering parameter of the original-unweighted 1D sinusoidal EBG filter has the non-negligible ripple, the ripple level is significantly improved using weighted techniques, and the bandwidth becomes wider through weighting. The rejection depth can be tuned by changing the weighted amplitude factor.

For the rejection depth of S21 = 32 dB, the amplitude factor is Rmax,G = 1.7 for Gaussian weighting functions, Rmax,Hm is 1.9 for Hamming weighting, Rmax,Hn is 2.0 for Hanning weighting, and Rmax,B is 2.5 for Blackman weighting. The simulated S-parameters of weighted EBG band-stop filters with a rejection depth of 32 dB are shown in Fig. 2. It is observed that weighted microstrip filters with various function forms show lower sidelobes and more flat ripple than original-unweighted sinusoidal structures. The weighted band-stop filter reduced the S11 sidelobe level, which implies an excellent transmission in the passband. Furthermore, the bandwidth of filters is expanded.

FIG. 2.

Simulated S11 (a) and S21 (b) of weighted EBG stop-band filters with different amplitude factors and the original EBG structure. Blackman function with 2.5-times amplitude factor, Gaussian function with 1.7-times amplitude factor, Hamming function with 1.9-times factor, and Hanning with 2.0-times factor.

FIG. 2.

Simulated S11 (a) and S21 (b) of weighted EBG stop-band filters with different amplitude factors and the original EBG structure. Blackman function with 2.5-times amplitude factor, Gaussian function with 1.7-times amplitude factor, Hamming function with 1.9-times factor, and Hanning with 2.0-times factor.

Close modal

The rejection depth is increased by increasing the amplitude factor of the weighting function. The simulated S21 parameters of the Gaussian-weighted band-stop filter are shown in Fig. 3: when the rejection depth is S21 = 17.5 dB, the amplitude factor of Gaussian weighting functions is Rmax,G = 1.0, and when the rejection depth is S21 = 30.5 dB, the amplitude factor of Gaussian weighting functions is Rmax,G = 1.6; and when S21 is 32 dB, Rmax,G is 1.7, and when S21 is 35 dB, Rmax,G is 1.8.

FIG. 3.

Simulated S-parameters of Gaussian-weighted EBG stop-band filters with different amplitude factors.

FIG. 3.

Simulated S-parameters of Gaussian-weighted EBG stop-band filters with different amplitude factors.

Close modal

Several weighted 1D EBG band-stop filters, including Gaussian, Hamming, Hanning, and Blackman weighting with different amplitude factors Rmax = 1.0, 1.6, 1.7, 1.8, 2.0, and 2.5, were fabricated. The substrate is F4B-2 with a dielectric constant of 3.5. The thicknesses of copper and dielectric are 35 µm and 1.5 mm, respectively. The impedance matching of 50 Ω is achieved by appropriately designing metallic strip dimensions at both ports. The S-parameters of these several filters are measured with an Agilent E5071C Vector Network Analyzer.

The fabricated Gaussian-weighted 1D EBG Microstrip structure with 1.7-times amplitude factor is depicted in Fig. 4. The yellow and gray parts in Fig. 3 stand for copper and dielectric substrate, respectively. Figure 5 shows the simulated and measured S-parameters of the fabricated Gaussian-weighted EBG structure with 1.7-times amplitude factor and original sinusoidal EBG structure. The simulated and experimental results are roughly identical, and the S-parameters of simulation and experiment have a slight difference from 5 to 7 GHz. This reason will be reflected more obviously at high frequency. The Gaussian-weighted 1D EBG filter demonstrates the advantages of large bandwidth, smooth passband, suppressed sidelobe, and deep rejection. The measurement results have a close match with simulation results, and consequently, the weighted design concept based on coupling-mode theory has effectively improved filter performances for the sinusoidal microstrip band-stop filter.

FIG. 4.

Photographs of the fabricated Gaussian-weighted EBG microstrip structure.

FIG. 4.

Photographs of the fabricated Gaussian-weighted EBG microstrip structure.

Close modal
FIG. 5.

S11 (a) and S21 (b) of the Gaussian-weighted 1D EBG structure with 1.7-times amplitude factor and original sinusoidal EBG structure.

FIG. 5.

S11 (a) and S21 (b) of the Gaussian-weighted 1D EBG structure with 1.7-times amplitude factor and original sinusoidal EBG structure.

Close modal

In this paper, the weighted techniques are introduced for the 1D sinusoidal microstrip band-stop filter. The passband ripple and reflected sidelobes of the proposed 1D weighted EBG structures are improved using weighting windows, and the rejection depth of the filter is tuned by changing the amplitude factor of weighting functions. Therefore, compared with the original-unweighted sinusoidal microstrip band-stop filter, the bandwidth of weighted filters is expanded. In a word, the filtering functionality of the 1D weighted EBG structure is flexibly controlled and effectively improved using weighting windows with various amplitude factors.

This research was partially supported by the Natural Science Basic Research Program of Shaanxi, China (Program No. 2020JQ-834).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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