Half-Elliptical Resonator Lowpass Filter with a Wide Stopband for Low Band 5G Communication Systems

Abstract

In this paper, a lowpass filter is designed using half elliptical resonators with a wide stopband. New formulas are presented to achieve a circuit model for the half elliptical resonators used in this work. Additionally, the transfer function and transmission zero equations are used to adjust the frequency of the transmission zeros of the filter. The cut-off frequency of the lowpass filter is 1.26 GHz with a sufficiently large stopband, extending from 1.48 GHz to 20 GHz. The proposed filter’s figure of merit is 62,520, demonstrating its outperformance compared to the state of the art. The filter is implemented on a RT-5880 substrate with a constant dielectric of 2.2, thickness of 31 mil and loss tangent of 0.0009. The LPF was fabricated and tested, showing good agreement between the simulated and measured results.

1. Introduction

Lowpass filters (LPFs) are one of the highly demanded devices in the modern communication circuit. In [1], a compact LPF is proposed with distinguishing features, such as sharp roll-off and low insertion loss. Butterworth and Chebyshev microstrip filters are designed in [2] using stepped-impedance structures. A microstrip LPF with a sharp roll-off using the analytical method is presented in [3]. A compact LPF is presented in [4] based on defected ground structure; however, this method has a complex fabrication process, which could be restrictive in many applications [5]. In [6], an ultra-wide stopband LPF is designed with a sharp response, but this filter has a complicated structure. Stepped-impedance structures, open tubs and radial-shaped resonators have widely been used in designing microstrip filters, particularly lowpass filters with a large stopband [7,8,9]. Square split-ring resonators are used to shape a small microstrip LPF with a new structure in [10], but this filter suffers from high insertion loss in pass-band. An electromagnetic band-gap structure has improved the efficiency of the applied lowpass filter in [11]. Unfortunately, this method has a complex fabrication process. Other resonators, such as U- and T-shaped resonators, and a complementary square split ring resonator were used in [12,13,14,15]. T-shaped and circular-shaped resonators are used in [13] to design a lowpass filter to achieve a sharp roll-off and extended stopband, where work, equivalent circuits and equations of transmission zeros (TZs) were presented; however, the size of the paper is undesirably large. The T-shaped resonators were utilized in [16,17] to present compact lowpass filters, which were relatively large. In [18], a modified U-shaped resonator is applied to create a compact LPF, but the figure of merit (FOM) parameter for this LPF is lower than the state of the art. In [19], a compact LPF was introduced using compact microstrip resonator cells. These cells present a wide suppression level, but insertion loss in passband for this work is poor.

In [20], lumped components (L and C) were used to reduce the circuit size and provide a wide stopband; however, the application of lumped components increased the complexity of the filter, and lumped elements cannot be used for high frequencies above 24 GHz.

Furthermore, neural networks, which are useful tools in solving real-life engineering problems [21,22,23,24,25,26,27,28], have been recently used to model the passive circuit, like LPF. However, designing microwave circuits based on neural networks is not a straightforward approach, which is more suitable for device modelling. In [29], open stubs with bent structures were used to provide a wide stopband, where several stubs were needed. The relative stopband of the filter is 1.72. Moreover, the proposed LPF is a suitable device for low-band 5G applications.

The 5G spectrum provides remarkably faster mobile speed and lower delay compared with previous generations. The 5G spectrum has low-band, mid-band and high-band frequency bands. Low-band, which is also called sub 1 GHz, is a frequency band below 1 GHz. The proposed LPF correctly passes the sub 1 GHz signals and suppresses unwanted harmonics at higher frequency bands.

2. LPF Design Process

The design procedure of the proposed low pass filter is illustrated in Figure 1 through 7 steps. The proposed LPF is designed based on the half-elliptical shaped resonator. In the first three steps, the half-elliptical shaped resonator, its corresponding LC equivalent (LCE) model and the equation of its created transmission zero are obtained. In step 1, an LC equivalent circuit with the desired response is presented. The proposed LC model creates a transmission zero at the desired frequency. In step 2, with Equation (14), the location of the transmission zero can be obtained analytically. In step 3, the realization of the proposed half-elliptical shaped resonator is presented with a microstrip line. In step 4, the validity of these three steps is investigated. If the EM simulation of the extracted microstrip resonator and schematic simulation of the LC model are in good agreement, this step passes. In step 5, another two half-elliptical shaped resonators are designed in a similar way. In step 6, two suppressing cell resonators are designed. In step 7, finally, the proposed low pass filter is shaped based on the resonators designed in the previous steps.

In this design, firstly, dual half-elliptical resonators are used to achieve a sharp transition band, and meandered lines are used for size reduction. Secondly, a half-elliptical resonator is used to generate a transmission zero to achieve a wider stopband. At the end, two suppression cells are added to attain a wider stopband.

2.1. Initial Low-Pass Filter

Figure 2 illustrates the layout of the initial LPF. This filter consists of five resonators, one half-elliptical resonator in the middle, two elliptical resonators on both sides and two rectangular resonators, which are used as attenuator cells. The dimensions of all applied cells will be described in separate sections. Microstrip technology was used for implementation of the proposed filter. This technology has several advantages, such as low fabrication cost, planar configuration and ease of integration with other Radio Frequency (RF) components; hence, it has been widely used in microwave component designs, including filters [30,31], metasurfaces [32,33,34], power dividers [35,36,37], phase shifters [38] and antennas [39,40,41,42]. The LCE circuit is used to locate the transmission zeros (TZs) of the resonator. Each resonator generates a transmission zero, which uses a transfer function (TF), and a specific LCE circuit can be applied.

2.2. The Half-Elliptical Shaped Resonator Design

The layout of the half-elliptical shaped resonator is demonstrated in Figure 3a. The response of this resonator is shown in Figure 3b. This resonator creates a TZ at 2.5 GHz. To increase the width of the stopband and eliminate more harmonics, this zero must be located at 3.5 GHz. An LC equivalent circuit can be used to set this transmission zero.

The half-elliptical shaped resonator and the rectangular resonator are shown in Figure 4. A new length “L” is calculated in Equation (3) and used to calculate the LCE circuit of the half-elliptical resonator. The “a” and “b” are the diameters of the half-elliptical shape. The semi-elliptical area is calculated in Equation (1), and the area of the rectangular shape is calculated in Equation (2). The new “L” is calculated in Equation (3) and used in Equations (4) and (5). The LC equivalent circuit of the semi-circular resonator is calculated in Equations (6) and (7).

$${\mathrm{area} }_{\mathrm{half}-\mathrm{elliptical}}=\frac{\mathsf{\pi }\mathrm{ab}}{2}$$

(1)

area _rectangular = L × 2a

(2)

$$\mathrm{L}=\frac{\mathsf{\pi }\mathrm{b}}{4}$$

(3)

$${\mathrm{L}}_{\mathrm{s}}=\frac{1}{2\mathsf{\pi }\mathrm{f}}\times {\mathrm{Z}}_{\mathrm{s}}\times \mathrm{Sin}\left(\frac{2\mathsf{\pi }\mathrm{L}}{{\mathsf{\lambda }}_{\mathrm{g}}}\right)$$

(4)

$${\mathrm{C}}_{\mathrm{s}}=\frac{1}{2\mathsf{\pi }\mathrm{f}}\times \frac{1}{{ \mathrm{Z}}_{\mathrm{s}}}\times \mathrm{Tang}\left(\frac{\mathsf{\pi }\mathrm{L}}{{\mathsf{\lambda }}_{\mathrm{g}}}\right)$$

(5)

$${\mathrm{L}}_{\mathrm{s}}=\frac{1}{2\mathsf{\pi }\mathrm{f}}\times {\mathrm{Z}}_{\mathrm{s}}\times \mathrm{Sin}\left(\frac{{\mathsf{\pi }}^2\mathrm{b}}{2{\mathsf{\lambda }}_{\mathrm{g}}}\right)$$

(6)

$${\mathrm{C}}_{\mathrm{s}}=\frac{1}{2\mathsf{\pi }\mathrm{f}}\times \frac{1}{{ \mathrm{Z}}_{\mathrm{s}}}\times \mathrm{Tang}\left(\frac{{\mathsf{\pi }}^2\mathrm{b}}{4{\mathsf{\lambda }}_{\mathrm{g}}}\right)$$

(7)

$$\mathrm{c}=\frac{\mathsf{\pi }\mathrm{b}}{4}$$

(8)

Equations (4) and (5) can be used to obtain the LC equivalent circuit. However, to obtain the equivalent half-elliptical circuit, a new formula must be obtained with the help of the half-elliptical resonator area and the simple rectangular resonator area, as shown in Equations (6) and (7).

According to Equations (4)–(7), the LCE circuit of the half-circular resonator is extracted and illustrated in Figure 5a. Different approaches to the extraction of equivalent circuits from microstrip filters are explained in [43,44,45,46]. The comparison of its response with the EM simulation results is shown in Figure 5b. As seen, this resonator produces a transmission zero (TZ) at 2.5 GHz. To tune this TZ at 3.5 GHz, the equation of TZ should be extracted. The transfer function (TF) can be used to obtain the equation of TZ.

To calculate the transfer function, the LC-equivalent circuit is divided to three parts, as shown in Figure 5a. In part 1, C_e3 and L_e3 are the series and corresponding impedance of part 1, calculated in Equation (9).

$${\mathrm{Z}}_{\mathrm{part}1}=\frac{1+{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^2}{{\mathrm{C}}_{\mathrm{e}3}\mathrm{S}}$$

(9)

Part 1 is parallel with C_e2; therefore, the impedance of part 2 is calculated in Equation (10).

$${\mathrm{Z}}_{\mathrm{part}2}=\frac{{\mathrm{C}}_{\mathrm{e}2}\mathrm{S}+{\mathrm{C}}_{\mathrm{e}3}\mathrm{S}+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^3}{1+{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^2}$$

(10)

Part 2 is the series with L_e2; therefore, the impedance of part 3 is calculated in Equation (11).

$${\mathrm{Z}}_{\mathrm{part}3}=\frac{1+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}2}{\mathrm{S}}^2+{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}2}{\mathrm{S}}^2+{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^2+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^4}{\mathrm{s}\left({\mathrm{C}}_{\mathrm{e}2}+{\mathrm{C}}_{\mathrm{e}3}+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^2\right)}$$

(11)

Finally, part 3 is parallel with C_e1; therefore, the impedance of the node “m” is calculated in Equation (12).

$${\mathrm{Z}}_{\mathrm{m}}=\frac{1+({\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}2}+{ \mathrm{C}}_{\mathrm{e}3}({\mathrm{L}}_{\mathrm{e}2+}{\mathrm{L}}_{\mathrm{e}3})){\mathrm{S}}^2+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^4}{\mathrm{s}\left({\mathrm{C}}_{\mathrm{e}1}+{\mathrm{C}}_{\mathrm{e}2}+{\mathrm{C}}_{\mathrm{e}3}+({\mathrm{C}}_{\mathrm{e}1}\left({\mathrm{C}}_{\mathrm{e}2}+{\mathrm{C}}_{\mathrm{e}3}\right){\mathrm{L}}_{\mathrm{e}2}+\left({\mathrm{C}}_{\mathrm{e}1+}{\mathrm{C}}_{\mathrm{e}2}\right){\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}3}\right){\mathrm{S}}^2+{\mathrm{C}}_{\mathrm{e}1}{\mathrm{C}}_{\mathrm{e}2}{\mathrm{C}}_{\mathrm{e}3}{\mathrm{L}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}3}{\mathrm{S}}^4)}$$

(12)

The TF is shown in Equation (13), where parameter “Z_m” is defined in Equation (12). In order to obtain the equation of transmission zero, the numerator of TF should be set to zero, where the equation of TZ is penned in Equation (14). As can be seen in Equation (14), many elements are important in the equation of TZ. The inductor with a value of L_e2 has an important effect on the location of the created TZ. Figure 6 shows the relationship between the location of TZ and the value of the L_e2 inductor. As seen, if this value decreases, the frequency of the created TZ increases. The values of capacitances and inductances in Figure 5a are summarized in

Table 1. The obtained values of capacitances and inductances.

Parameters	Le1	Le2	Le3
Values(nH)	8.6	1.37	1.32
Parameters	Ce1	Ce2	Ce3
Values(pF)	0.255	0.335	1.7

.

$$\frac{{\mathrm{V}}_{\mathrm{o}}}{{\mathrm{V}}_{\mathrm{i}}}=\frac{{\mathrm{Z}}_{\mathrm{m}}\mathrm{r}}{\left(\mathrm{r}+{\mathrm{L}}_{\mathrm{e}1}\mathrm{s}+{\mathrm{L}}_{\mathrm{e}2}\mathrm{s}\right)\left(2{\mathrm{Z}}_{\mathrm{m}}+\mathrm{r}+{\mathrm{L}}_{\mathrm{e}1}\mathrm{s}+{\mathrm{L}}_{\mathrm{e}2}\mathrm{s}\right)}$$

(13)

where r is 50 ohms. The characteristic impedance and Z_m, are calculated in Equation (12).

$$\mathrm{TZ}=\frac{\sqrt{-\frac{1}{{\mathrm{C}}_{\mathrm{e}1}{\mathrm{L}}_{\mathrm{e}1}}-\frac{1}{{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}1}}-\frac{1}{{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}2}}-\frac{\sqrt{-4{\mathrm{C}}_{\mathrm{e}1}{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}1}{\mathrm{L}}_{\mathrm{e}2}+{\left({\mathrm{C}}_{\mathrm{e}1}{\mathrm{L}}_{\mathrm{e}1}+{\mathrm{C}}_{\mathrm{e}1}{\mathrm{L}}_{\mathrm{e}2}+{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}2}\right)}^2}}{{\mathrm{C}}_{\mathrm{e}1}{\mathrm{C}}_{\mathrm{e}2}{\mathrm{L}}_{\mathrm{e}1}{\mathrm{L}}_{\mathrm{e}2}}}}{\sqrt{2}}$$

(14)

Based on Figure 6, if L_e2 is 0.17 nH, the frequency of TZ is located near 3.5 GHz. The modified structure of this resonator is depicted in Figure 7a, and its frequency response is illustrated in Figure 7b.

2.3. Two Elliptical Resonators Design

Two elliptical resonators are used to achieve a very sharp response. For this purpose, the structure of these resonators is shown in Figure 8a, which shows two elliptical resonators. These two resonators can produce a suitable sharpness. The frequency responses of two elliptical resonators are demonstrated in Figure 8b.

2.4. Attenuator Resonator Design

In the previous two types of resonators, two transmission zeros were produced at low frequencies. In order to produce a TZ at higher frequencies, an attenuator needs to be used. In this design, a square-shaped attenuator is used. This square-shaped attenuator can create a TZ at a frequency of 11 GHz. The attenuator layout and its frequency response are depicted in Figure 9a,b.

2.5. Design of the Proposed Filter

In the proposed design, to reduce the occupied size, applied lines are bent. The structure of the proposed filter is depicted in Figure 10a, for which all dimensions were previously obtained, and its frequency response is depicted in Figure 10b. The proposed LPF has a cut-off frequency of 1.26 GHz and provides a wide stopband from 1.48 GHz up to 20 GHz with more than 24 dB of attenuation level.

3. Implementation and Measurement Results

Figure 11 shows a photo of the fabricated filter, along with its frequency response for both measurements and the simulation results. The measurement shows that the cut-off frequency is 1.26 GHz. As seen in this figure, the proposed LPF has a passband insertion loss (IL) of less than 0.3 dB and a return loss (RL) higher than 12 dB.

The roll-off rate (ζ) parameter, which shows the sharpness of S₂₁ in the transition band, is defined in (8) as follows:

$$\mathsf{\zeta }=\frac{{\mathsf{\alpha }}_{\max }-{\mathsf{\alpha }}_{\min }}{{\mathrm{f}}_{\mathrm{s}}-{\mathrm{f}}_{\mathrm{c}}} (\mathrm{dB}/\mathrm{GHz}),$$

(15)

where α_max and α_min are 40 dB and 3 dB attenuation points, while f_s and f_c are the corresponding frequencies of these points. In the proposed LPF, the transition band is from 1.26 GHz to 1.62 GHz with attenuation levels of 3 dB and 40 dB, respectively. As a result, the roll-off rate (ζ) parameter of the fabricated filter is 103.9 dB/GHz.

Another important parameter is relative stopband bandwidth (RSB), which is defined in Equation (16) as follows:

$$\mathrm{RSB}=\frac{\mathrm{stopband} \mathrm{bandwidth}}{\mathrm{stopband} \mathrm{center} \mathrm{frequency}}$$

(16)

In the proposed LPF, the stopband ranges from 1.48 to 20 GHz with the attenuation level of 24 dB. Therefore, the stopband bandwidth is 18.52 GHz and the center frequency of the stopband is 10.74 GHz. As a result, the RSB is equal to 1.72.

The suppression factor (SF) parameter is defined in Equation (17) as follows:

$$\mathrm{SF}=\frac{\mathrm{Rejection} \mathrm{level} \mathrm{in} \mathrm{stopband} }{10}$$

(17)

As mentioned in the proposed LPF, the stopband ranges from 1.48 to 20 GHz with the attenuation level of 24 dB. Therefore, the SF is 2.4.

The cut-off frequency of the LPF (fc) and its occupied size are inversely proportional. The filter with the higher cut-off frequency has a smaller size and vice versa. Therefore, the normalized circuit size (NCS) parameter is defined in Equation (18) to have a fair comparison. The relative device size is 0.126 λ_g × 0.055 λ_g.

$$\mathrm{NCS}=\frac{\mathrm{physical} \mathrm{size} \left(\mathrm{length}\times \mathrm{width}\right)}{{\mathsf{\lambda }}_{\mathrm{g}}{}^2},$$

(18)

where λ_g is the guided wavelength. The figure of merit (FOM) parameter has been widely used in the literature and shown in Equation (19) as follows:

$$\mathrm{FOM}=\frac{\mathsf{\zeta } \times \mathrm{RSB} \times \mathrm{SF} }{\mathrm{NCS} }$$

(19)

The calculated figure of merit (FOM) parameter of the proposed filter is 62,520, which is considerably high for this class of filter compared with other published works. The proposed filter is compared with other published filters in

Table 2. Comparison between the proposed filter and some published filters.

Ref	fc (GHz)	ζ (dB/GHz)	NCS (λg2)	RSB	SF	RL	IL	FOM	Substrate
[16]	1.89	139	0.0177	1.69	2.5	10.3	1.8	33,179	Rogers
[18]	2.35	135	0.032	1.64	2.2	10.6	0.6	18,562	RT/Duroid
[46]	2.4	92.5	0.037	1.35	3	10	0.25	10,125	RT/Duroid
[47]	1.96	104	0.0228	1.8	2	12	0.6	18,063	Rogers
[48]	1.47	85.6	0.0076	1.68	2.2	12	0.4	41,628	RT/Duroid
[49]	2.44	67.27	0.038	1.39	2.2	14.8	0.6	5414	FR4
[50]	3.01	58.6	0.0242	1.43	2.5	20	0.3	865.5	RT/Duroid
[51]	5.55	84	0.0857	0.66	1.5	20	0.7	116.5	FR4
[52]	2.11	100	0.032	1.6	1.8	12	0.7	9000	FR4
[53]	1.07	75	0.0088	1.66	2	20	0.4	2804	DiClad
[54]	1	77	0.0117	1.74	2.4	20	0.3	27,192	Rogers
[55]	2.49	37.5	0.0378	1.62	1.5	7.6	0.8	2410	Rogers
[56]	1.66	171	0.0176	1.75	2	12	0.5	34,000	Rogers
This Work	1.26	103.9	0.00693	1.73	2.4	12	0.3	62,520	RT/Duroid

.

Table 2 shows that the proposed LPF has the smallest NCS and the highest FOM parameters compared with other related works.

4. Conclusions

A new microstrip lowpass filter with a small occupied size (0.126 λ_g × 0.055 λ_g) and ultra-wide stopband (1.48 to 20 GHz) is proposed. The proposed filter has an attenuation level higher than 24 dB in stopband and a very sharp response in the transition band with a sharp roll-off rate higher than 103 dB/GHz. With these specifications, the implemented device can be used in modern wireless systems.