In this article, a compact microstrip lowpass filter (LPF) using elliptical shaped resonators with ultra-wide stopband is rendered. In this respect, LC equal structures of the elliptical shaped resonators are calculated based on the formula of circumference. In addition, to calculate transmission zeros of the presented elliptical shaped resonator, the LC equal structure and its output to input ratio are employed. The proposed LPF has a −3 dB cut-off frequency at 1.50 GHz and the stopband bandwidth of the designed filter is about 13fc, which refers to its ultra-wide stopband. The occupied circuit size of the presented filter is 0.151λ g × 0.044λ g (λ g is the guided wavelength at 1.50 GHz). The designed filter is fabricated on RT/Duroid 5880 substrate. The results of the fabricated and designed filter have clearly demonstrated that not only has the proposed LPF shown a suitable agreement between measured and simulated S-parameters, but also an appropriate stopband bandwidth.
Compact microstrip LPFs with ultra-wide stopband are extremely desirable for wireless and mobile communication systems to suppress unwanted high frequency harmonics. These systems need the LPF circuits with high performance. Therefore, LPFs with good frequency features are very noticeable in many communication systems [1, 2]. Several studies have been done to implement and analyze LPFs and microwave circuits with high performance , , , , , , , , , , , , , , , , , , , , , , , , , , , , . One method, which is used in LPFs implementation, is Defected Ground Structure (DGS) to obtain high stopband suppression and size reduction [3, 4]. Another method is using stub-loaded resonators such as a compact LPF with a wide stopband, which is presented in . Using coupled rhombic stubs, a compact and highly selective LPF with a wide stopband is presented in . Authors in  applied hexangular shaped resonators for designing a LPF with high performance. Another LPF with compact size and sharp roll-off using modified Complementary Split-Ring Resonator (CSRR) was presented in . A compact harmonic suppressed microstrip structure is presented in , which consists of stepped impedance lines and symmetric modified semicircular patches. The dual-plane structure is another method that is used to design compact LPFs as mentioned in . Another type of LPF is presented in , which is called multi stub LPF, in this structure the open stubs are used instead of impedance lines. A compact microstrip LPF with ultra-wide stopband is designed in , which uses an analyzed triangular shaped resonator in its design. Using a lattice-shaped resonator, a compact LPF with flat group delay with a wide rejection band and sharp roll-off is presented in . Another LPF with sharp roll-off and ultra-wide stopband using circular sectors and cross-coupled structures is designed in . A very applicable procedure for designing a lowpass filter is presented in , which some polygonal structures are embedded in its design and its analysis has been done based on transfer function evaluation. In , using LPF design, a dual-wideband bandpass filter with a wide upper stopband is presented, which uses a short stepped-impedance stub-loaded resonator in its LPF design. Another LPF with compact and ultra-wide stopband is designed in , which uses a narrow width microstrip line with high impedance and three multimode resonators in its structure. Using fractal structures and hairpin units are other methods to design the LPFs, such as , , . Authors in  applied non-uniform radial stubs resonators to design another compact microstrip LPF with a wide stopband. A new method to enhance the LPFs performances is using neural network analysis, in , another LPF is designed and fabricated using multiple open stubs and stepped impedance resonators, which its features have been improved utilizing the mentioned method. Another lowpass filter with a sharp rejection band using coupled T-shaped, elliptical and radial resonators is presented in . Using four high-low impedance patches and four radial stub structures, a lowpass filter with efficient performances is designed in , which has a high relative stopband bandwidth. Based on inductor-capacitor equivalent circuit analysis and using some series resonators with meandered lines, another LPF is designed in . Another method to design and analyze the filters and other microstrip devices is using even and odd mode technique, [26, 27]. Recently, LPFs are used to develop other microwave devices for example , , , .
In this paper, a microstrip LPF with considerable features, such as ultra-wide stopband, compact size and simple design with a return loss better than 29 dB is presented. The proposed LPF with −3 dB cut-off frequency at 1.50 GHz is designed using elliptical resonators. In addition to the good features of the suggested design, a new formula is presented to calculate the LC equivalent structure of the elliptical structures based on its geometric circumference, which is a new technique to calculate the LC elements.
2 Filter design
The filter design consists of creating some elliptical resonators and computing their output to input ratios (transfer functions), to analyze their behavior. The elliptical resonators are selected due to their suitable frequency responses, and by applying these types of resonators in different sizes a filter with sharp transition band and wide stopband can be designed. Also, based on its circumference a new formula for calculating LC equivalent circuit can be computed and rendered.
2.1 First elliptical resonator (FER)
The topology and the LC equivalent structure of the first elliptical resonator are shown in Figure 1(a) and (b), respectively. Also, the Electro Magnetic (EM) and LC frequency responses of the FER are illustrated in Figure 1(c), up to 10 GHz. The first transmission zero of this structure occurs at 9.30 GHz with an attenuation level better than 48 dB, which is suitable to design a wide stopband LPFs. Dimensions of this resonator are (All in millimeters): L1 = 5.7, L2 = 0.51, W1 = 8.56, W2 = 2, W3 = 0.1.
The values of capacitances (C x ) and inductances (L x ) are computed using open ends and high-low impedance lossless lines formulas, . Equations (1) and (2) express these formulas, where Z s is the characteristic impedance of the lines and λ g in Eqs. (1) and (2) is calculated by the width of lines  (Note: λ g is the calculated guided wavelength at 1.50 GHz).
Equation (3) is obtained based on the geometric circumference of the elliptical shaped structure. The elliptical resonator is formed using half of an elliptical shaped patch. The Wnew in Eq. (3) is approximately equal to 1/2 of a semi circumference of an elliptical shaped patch. Therefore, the circumference of an elliptical shaped cell have to be multiplied by 1/4, but the 1/4 of the elliptical circumference has a small error in comparison with EM simulation. To decrease this error, a new formula is computed, based on linearization of error. So, if W/L becomes equal to 1.12, the error will be equal to approximately zero. Therefore, 3L/W and 2L/W in the first and second lines of Eq. (3) are calculated for decreasing the error. The length (L) and width (W) in Eqs. (1) and (2) are replaced with Lnew and Wnew which are given in Eqs. (3) and (4) for achieving the values of the LC equivalent structure of the elliptical resonators. Then, using open ends formulas , inductances and capacitances are computed. Calculated values for the LC equivalent structure of the FER are as follows. These amounts are shown by C fi (pF) and L fi (nH), parameters (Note: the “f” index is related to FER design and i = 1, 2, …).
Lf1, Lf2, Lf3 and Lf4 (Eq. (1)) and Cf1 (Eq. (2)) are denoted to the inductances and capacitance of high impedance lossless lines. Cf2 is the summation of the capacitances of the resonator and high impedance lossless line (Eqs. (2)–(4)). Cf3 denotes the capacitances of the resonator and its open-end effect (formulas of gap , Eqs. (3) and (4)). Lf3 (Eqs. (3) and (4)) refers to the inductance of the investigated resonator. These LC equivalent elements are assigned and shown in Figure 1(d).
The output to input ratio of the investigated elliptical resonator is calculated in Eq. (5). The locations of transmission zeros can be determined by the output to input ratio. The designed structure can generate a deep transmission zero at 9.30 GHz, which is calculated in Eq. (6). It is shown in Figure 1(c) by Tz1.
For example, Tz1 can be changed by Lf3 and Cf2 based on Eq. (6), where other inductances and capacitances values are fixed. If Cf2 is assumed as 0.94, 0.97 and 1.13 pF, and Lf3 is assumed as 0.35, 0.38 and 0.42 nH, then Tz1 will be 10.55, 9.30 and 8.39 GHz, respectively. These calculated parameters are summarized in Table 1. For having mentioned transmission zeros, L1 can be calculated about 5, 5.7 and 6.4 mm, respectively, based on Eqs. (1) and (2).
where: ( )
|TZ1-FER = 10.55 GHz||TZ1-FER = 9.30 GHz||TZ1-FER = 8.39 GHz|
|C f2||0.94 pF||C f2||0.97 pF||C f2||1.13 pF|
|L f3||0.35 nH||L f3||0.38 nH||L f3||0.42 nH|
2.2 Second elliptical resonator (SER)
As discussed, the first transmission zero and pole of the FER are occur at 9.30 GHz and 1.21 GHz, respectively, but this transmission zero and transmission pole are not near to each other, so the transition band is not sharp in this configuration. To improve the sharpness of the transition band, the SER is designed as seen in Figure 2(a). The LC equivalent structure of the SER is shown in Figure 2(b). The results of the EM and LC simulations are shown in Figure 2(c), up to 10 GHz.
The configuration of the SER generates a transmission zero at 1.97 GHz with the attenuation level better than 48 dB. The structure has a transmission pole at 1.08 GHz, too. So the transmission pole and zero are happen approximately near to each other; therefore, these double and smaller elliptical shaped structures helps to move the first LPF’s transmission zero to the lower frequencies and leads to an increase in sharpness of the final LPF’s transition band.
SER’s dimensions are (All in millimeters): L3 = 5.45, L4 = 6.4, L5 = 2, L6 = 0.19, L7 = 2.54, L8 = 16.7, W4 = 4.17, W5 = 0.1, W6 = 2.44, W7 = 1.63. The LC equal values of the SER are shown by C si (pF) and L si (nH), parameters (i = 1, 2, 3 … );
The output to input ratio of the LC equivalent structure for the SER, as shown in Figure 2(b), is calculated in Eq. (7). Using this ratio leads to determine the transmission zero locations (Note: r = 50 Ω).
3 Prototype lowpass filter
The simulation results of the FER and SER are shown in Figure 3(a), up to 20 GHz. As seen, the FER generates three transmission zeros and the SER can create two transmission zeros, too. In order to design a prototype LPF with a sharp transition band and wide stopband bandwidth, both first and second elliptical resonators are combined and a prototype LPF with five main transmission zeros plus some coupling effects is formed. This designed structure, and its simulation results are shown in Figure 3(b), while the transmission zeros are assigned by T zmi (i = 1, 2, …, 5). As seen in Figure 3(b), this configuration has not a good suppression level in the stopband.
To increase the level of suppression in “a, b and c” points, two attenuator cells are located in series with the prototype LPF. Eventually, the final LPF with −3 dB cut-off frequency at 1.50 GHz is designed. The total equivalent LC circuit is shown in Figure 4(a). As seen in Figure 4(b), the complementary dimensions of the proposed LPF are (All in millimeters): L9 = 1.34, L10 = 4.5, L11 = 0.2, L12 = 7.35, W8 = 1. Also, the LPF LC equal values are (Units: C ti , pF; L ti , nH, i = 1, 2, 3 …);
Lt1 = 0.89, Lt2 = 0.13, Lt3 = 4.86, Lt4 = 4.23, Lt5 = 1.39, Lt6 = 0.52, Lt7 = 0.08, Lt8 = 0.39, Lt9 = 1.12, Ct1 = 0.2, Ct2 = 0.19, Ct3 = 0.23, Ct4 = 0.11, Ct5 = 0.45, Ct6 = 0.41, Ct7 = 0.97, Ct8 = 0.85, Ct9 = 0.18 (The values of the inductances and capacitances are calculated using Eqs. (1)–(4)). The results of the EM and LC simulations are shown in Figure 4(b), up to 10 GHz.
4 Simulation and experimental results
The proposed LPF is designed and fabricated on an RT/Duroid 5880 substrate. The substrate details are mentioned in Table 2.
|Relative dielectric constant||Loss tangent||Thickness (mils)|
The photograph and S-parameters of the fabricated and simulated LPF are shown in Figure 5(a) and (b), respectively. The measured results are obtained by the Agilent Network Analyser N5230A. The proposed LPF’s cut-off frequency is 1.50 GHz, the stopband bandwidth is expanded from 1.84 up to 21.39 GHz with 20 dB attenuation level, and this stopband region is about 13fc. In the passband region, a suppression level of 29.7 dB is obtained. The size of the LPF circuit is only about 22 × 6.45 mm2. The designed filter has a high Figure of Merit (FOM) equal to 42,759. For comparison, Table 3 shows a summary of key LPF performance metrics , for current designs.
|Ref.||RO (ξ)||RSB||SF||NCS (λ g 2)||AF||FOM|
|||48.5||1.49||2.5||0.160 × 0.080||1||14,062|
|||36||0.96||1.2||0.350 × 0.200||1||600|
|||57.8||1.61||3.5||0.120 × 0.100||1||27,142|
|||84.7||1.51||2||0.143 × 0.156||1||11,625|
|||188.8||0.92||2||0.200 × 0.180||1||9681|
|||25.3||1.34||2||0.186 × 0.101||1||3626|
|||74||1.74||2.4||0.090 × 0.136||1||25,274|
|||100||1.58||2||0.670 × 0.110||1||4288|
|||104||1.80||2.5||0.189 × 0.121||1||22,510|
|||52.8||1.52||2||0.081 × 0.113||1||17,638|
|This work||84||1.69||2||0.151 × 0.044||1||42,759|
RO means roll-off rate or selectivity according to 3 and 40 dB suppressing points . RSB, relative stopband bandwidth; SF, suppression factor; NCS, normalized circuit size; AF, architecture factor; FOM, figure of merit.
In this paper, a simple and compact microstrip LPF with an ultra-wide stopband based on elliptical resonators is designed, fabricated and measured. The presented LPF has a −3 dB cut-off frequency at 1.50 GHz. The proposed filter has many desirable features, each of which depicts proper characteristics for a LPF theoretically and practically, such as compact size, return loss better than 29 dB, and ultra-wide stopband from 1.84 GHz to 21.39 GHz. With all these good features, the proposed LPF is recommended for modern communications systems.
Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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