Abstract
A band-stop filter with three different negative group delay functions in the passband, namely, band pass, high pass and low pass, is proposed, which has small insertion loss. The capacitance, inductance, and resistance meet different conditions, and the circuit can realize three different negative group delay characteristics. The theoretical calculation and equation derivation are given. A band-stop filter with negative group delay function is fabricated, and the measured results are basically consistent with the simulation results. The correctness of the design is verified.
1. Introduction
Band-stop filters (BSFs) are used for blocking undesired signals in RF/microwave systems [1]. At present, much work has been done on band-stop filters [2–15]. Extended passband band-stop filter cascade with continuous 0.85–6.6 GHz coverage is presented in [2]. Compact wideband band-stop filters with five transmission zeros are depicted in [3].
The research on filters is presented in a recently published work. Compact low-pass and dual-band band-pass filters with controllable transmission zero/center frequency/passband bandwidth are proposed in [16]. The position of transmission zeros/center frequencies and 3 dB bandwidth characteristics can be separately controlled by varying the physical parameter of the designed filter structure. In [17], a compact D-CRLH resonator for low-pass filters with wide rejection band, high roll-off, and transmission zeros is proposed. The filter has wide rejection band bandwidth with three transmission zeros (TZs). The filter bandwidth and TZ frequencies are controlled by the D-CRLH element values, and the filter has minimum insertion loss in passband, high roll-off rate, and good figure of merit. In [18], miniaturized quad-band filters with improved selectivity using split ring resonators and metallic strips are proposed. The operating frequency of the proposed filter structure can be controlled with the variation in the length ratio of resonators.
As the performance index of filters, group delay affects the performance of communication systems [19,20]. The methods to reduce the group delay based on the all-pass network are designed in [21, 22]. Negative group delay (NGD) technology has attracted attention for reducing the influence of group delay. An equalization technique based on negative group delay is proposed in [23, 24], which can reconstruct the signals changed by time delay. The NGD circuits are realized by using series, shunt RLC resonators, or microwave regime. Although some active schemes of NGD circuits have been conceived, passive realizations have attracted more interest due to their higher simplicity [25]. Passive band-pass, low-pass, and high-pass NGD circuit topologies are presented in [26–28].
Negative group delay is gradually applied to filters. A class of NGD band-stop filter (BSF) circuits with input-reflectionless behavior are presented in [25], cancelling the undesired RF-power reflections that can adversely affect preceding active tags. In [29], reflective BSFs are designed with enhanced negative group delay bandwidth (NBW), where an additional impedance transformer is required to obtain good in-band matching. An absorptive band-stop filter is proposed and synthesized with prescribed NGD and NBW in [30].
At present, band-stop filters with negative group delay function generally achieve band-pass negative group delay, the insertion loss is large, and the signal has great attenuation. The NGD property is usually attained at the expense of significant signal-attenuation levels in lossy passive realizations of NGD circuits. In this paper, the LC band-stop filter with negative group delay of band pass, high pass and low pass is designed, and the insertion loss in the frequency range of negative group delay is improved.
This paper consists of six main parts. Section 2 introduces the structure of this circuit. The equations of band-stop filters satisfying different negative group delays are analyzed. The influence of each element value on the band-pass negative group delay is given. Section 3 discusses high-pass negative group delay in the band-stop filter's passband. In Section 4, low-pass negative group delay in the band-stop filter's passband is analyzed. In Section 5, the theory, simulation, and measurement are carried out. The final part of the paper is conclusion.
2. Band-Stop Filter Design with Negative Group Delay Function
The capacitor C1 and inductor L1 are connected in parallel to form a simple band-stop filter with an impedance value of Z1, as shown in Figure 1. The impedance value Z1 is given in equation (1). The transfer function S21(Z1) is given in equation (2), where Z0 is the characteristic impedance. The quality factor of the filter is . f0 is the resonant frequency of the filter. BW3 dB is the 3 dB bandwidth. The higher the Q value, the narrower the bandwidth.
(a)
(b)
When there are only L1 and C1 in Figure 1, the group delay is calculated as shown in equation (3). When ω > 0, L1 and C1 are taken at any value, the group delay of the circuit is always positive, and there is no negative group delay. That is, τ1(ω) > 0.
As shown in Figure 1, a negative group delay circuit with an impedance value of Z2 is added on the left side of the band-stop filter with impedance Z1. The impedance value Z2 is given in equation (4). The transfer function of the whole circuit is given in equation (5), and the group delay is given in equation (6). When ω > 0 and elements take specific values, negative group delay appears in this circuit. That is, τ2(ω) < 0 when different conditions are met in the passband. The transfer function of negative group delay band-stop filter is given in equation (7), and the group delay function is given in equation (8).When elements take different values, three negative group delay functions such as band pass, high pass, and low pass can be realized in the passband of the band-stop filter.
Compared with S21(Z1) and S21, the passband and stopband of the simple band-stop filter and negative group delay band-stop filter do not change. Only the attenuation of S21(Z1) and S21 is different. When L1 = 3.9 nH, C1 = 120 pF, L2 = 970 nH, C2 = 1.6 pF, the simulations are as shown in Figures 2 and 3.
When τ2(ω) = 0 and R1 = R2 = Z0 = 50 Ω, according to equation (6), two significant frequency values are calculated, which are, respectively, given in equations (9) and (10). f0 represents the frequency corresponding to the minimum value of negative group delay in equation (11). The circuit with impedance Z2 shown in Figure 1 is a band-pass negative group delay circuit if equation (12) is satisfied. It is a high-pass negative group delay circuit if equation (13) is satisfied. It is a low-pass negative group delay circuit if equation (14) is satisfied.
After adding the negative group delay circuit with the impedance of Z2 to the band-stop filter with the impedance of Z1, if and equations (12)–(14) are satisfied, respectively, the whole circuit can realize three different negative group delay band-stop filters of band pass, high pass, and low pass. The negative group delay band-stop filter is only attenuated by 1.9 dB compared with the original band-stop filter, but the negative group delay transmission of some signals in the passband is realized, and the delay performance of the filter is improved.
2.1. Band-Stop Filter with Band-Pass Negative Group Delay at Different L2 Values
When C1, C2, and L1 are fixed and R1 = R2 = Z0 = 50 Ω, the negative group delay circuit presents band-pass negative group delay. With the decrease in the inductance L2 in the negative group delay circuit, the minimum negative group delay in the passband of the band-stop filter increases and it satisfies equation (15). The larger the inductance L2 is, the smaller the negative group delay will be. The frequency of the minimum of negative group delay is about. The circuit appears as a bass-pass NGD according to the values of components in Table 1. The band-pass negative group delay and S21 at different values are shown in Figures 4 and 5, respectively.
Here,
2.2. Analyzing the Influence of C2 on Band-Stop Filters with Band-Pass Negative Group Delay
Compared with Table 1, the C2 value of the negative group delay circuit is changed, while the other element values remain unchanged. The performance indexes of negative group delay of the band-stop filter are shown in Table 2. In the passband of the band-stop filter, the minimum value of negative group delay changes little. Due to the change in C2 value, the frequency of minimum negative group delay changes. It can be seen from Tables 1 and 2 that there is negative group delay in the passband of the band-stop filter. In the negative group delay circuit, the influence of capacitor C2 on the group delay is small and the influence of inductance L2 on the minimum negative group delay is large. The resonant frequency of the band-stop filter is 233 MHz, and the frequency of the minimum negative group delay avoids the value near 233 MHz. At this time, the negative group delay cannot be generated in the passband of the band-stop filter. The band-pass negative group delay and S21 at different C2 are presented in Figures 6 and 7, respectively.
2.3. Analyzing the Influence of C1 and L1 on the Band-Stop Filter with Band-Pass Negative Group Delay
Compared with Table 2, changing the resonant frequency of the band-stop filter, that is, changing C1 and L1, and keeping C2, L2, R1, and R2 unchanged, the negative group delay value is basically unchanged. The data are depicted in Tables 3 and 4. At this time, only the resonant frequency of the band-stop filter changes and the negative group delay value in the passband and the corresponding negative group delay frequency remain unchanged.
The capacitor C1 affects the group delay value of the band-stop filter at the resonant frequency, as shown in Figure 8. However, the inductor L1 does not affect the group delay value of the band-stop filter at the resonant frequency, as shown in Figure 9. The characteristics of insertion loss and return loss with different C1 and L1 values are presented in Figures 10 and 11.
2.4. Analyzing the Influence of R1 on the Band-Stop Filter with Band-Pass Negative Group Delay
The resistance R1 has a great influence on the negative group delay. The frequency of the negative group delay in the passband of the band-stop filter is the resonant frequency of the negative group delay circuit. The data are shown in Table 5. The group delay and insertion loss characteristics at different R1 are shown in Figures 12 and 13.
2.5. Analyzing the Influence of R2 on the Band-Stop Filter with Band-Pass Negative Group Delay
When C1 = C2 = 120 pF, L1 = 20 nH, and L2 = 470 nH, R2 mainly affects the negative group delay bandwidth of the band-stop filter in the passband but has little effect on the minimum negative group delay, as depicted in Figure 14. With the increase in resistance R2, the negative group delay bandwidth of the band-stop filter in the passband increases, as shown in Tables 6. The characteristics of insertion loss are shown in Figure 15. At the minimum negative group delay, the loss is only 3.5 dB. The Figure of Merit (FOM) of designed filters is given in equation (16).When R2 = 200 Ω in Table 6, FOM = 0.0639.where τ(f0) represents the minimum negative group delay. BNGD represents the band of NGD. S21(f0) represents the insertion loss at frequency f0.
3. High-Pass Negative Group Delay in Band-Stop Filter Passband
According to the values of L1, L2, C1, and C2 in Table 7 and R1 = R2 = 50 Ω, the circuit presents a high-pass negative group delay circuit in band-stop filter passband. The condition for forming high-pass negative group delay in equation (13) is satisfied. The negative group delay decreases with the increase in capacitance while the inductance and resistance remain unchanged. The simulation results are shown in Figures 16 and 17.
4. Low-Pass Negative Group Delay in the Passband of the Band-Stop Filter
The circuit appears as a low-pass NGD circuit according to the values of components L1, C1, R1, and R2 in Table 8. According to equation (14), when the inductance decreases, the negative group delay increases. The cutoff frequency and the bandwidth increase. The simulation results are shown in Figures 18 and 19, from which it can be seen that the influence of inductance on the parameters of the low-pass negative group delay circuit.
5. Measurement and Discussion of Band-Pass NGD of Band-Stop Filter Circuits
When the element values are, respectively, C1 = C2 = 120 pF, L1 = 20 nH, L2 = 470 nH, R1 = R2 = 50 Ω, and Z0 = 50 Ω, the circuit is fabricated according to the element values. It is shown in Figure 20. The conditions of forming a band-pass negative group delay in the passband of the band-stop filter are satisfied. The theoretical, simulated, and measured waveforms are shown in Figures 21 and 22. Due to the influence of parasitic parameters in the parallel circuit, the simulation results deviate from the theoretical calculation. Compared with the simulation results, the group delay changes from positive to negative at the resonant frequency of the band-stop filter. The simulation results and the measured results are basically the same in the minimum negative group delay and the center frequency corresponding to the minimum negative group delay.
The comparison results of theoretical, simulation, and measurement performance indicators are shown in Table 9.
The comparison of the band-pass negative group delay in the passband of the band-stop filter with the reported literature is shown in Table 10. Compared with other band-stop filters, the designed band-stop filter with band-pass negative group delay operates at a lower frequency. Compared with [33], the negative group delay is reduced by 5.996 ns and the insertion loss is decreased by about 2.55 dB. Compared with [26], the insertion loss is decreased 19.15 dB. Compared with [27], the insertion loss is decreased by 19.1 dB. It can be seen from the comparison that this design has better insertion loss.
6. Conclusion
A band-stop filter with the function of negative group delay is proposed. When the values of elements are different, different types of negative group delay functions such as band pass, high pass, and low pass are realized in the passband of the band-stop filter. The band-pass-type negative group delay is realized in the band-stop filter’s passband, and the simulation results are basically consistent with the measured results. The theoretical minimum negative group delay in the passband of the filter is about −6.03 ns, which improves the delay performance of the communication system. The realization of negative group delay is always at the cost of signal attenuation. In the passband of the designed band-stop filter, the realization of negative group delay only has the insertion loss of 3.5 dB, while in many papers, it is greater than 20 dB attenuation, which improves the performance of insertion loss. The designed negative group delay band-stop filter has high practical value.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 61871417 and 51809030), the Liaoning Revitalization Talents Program (No. XLYC2007024), the Natural Science Foundation of Liaoning Province (Nos. 2019-MS-024 and 2020-MS-127), and the Fundamental Research Funds for the Central Universities (No. 3132021234).