A Novel Multifunctional Negative Group Delay Circuit for Realizing Band-Pass, High-Pass and Low-Pass

: A novel multifunctional negative group delay circuit is proposed. The circuit can realize three different negative group delay functions, including band-pass, high-pass and low-pass, which meet different conditions with capacitance, inductance and resistance. Analytical design equations are provided. The effects of different element values on the bandwidth, cut-off frequency and the minimum negative group delay of the circuit are analyzed. According to this design method, the negative group delay circuit is designed and fabricated. Simulation and measurement results are in agreement. It has good negative group delay characteristics. The feasibility of the design method is veriﬁed.


Introduction
Low latency is an important goal of modern communications. The group delay effect reduces the performance of the electronic circuit, limits the operating speed of the electronic system [1,2] and affects the performance of the communication system [3,4]. In order to eliminate the influence of group delay, many studies have been carried out. A group delay compensation technology is proposed in [5][6][7][8]. The synchronization technology based on memristive elements to maintain the integrity of the signal waveform is implemented in [9,10], but this technology will increase the power consumption of the system. A method to reduce the group delay based on the all-pass network is designed in [11,12]. This method can reduce the dispersion effect, but the reduction in the group delay is not obvious.
Negative group delay (NGD) technology has attracted attention for reducing the influence of group delay. Since the phenomenon of negative group delay is confirmed by experiments, scholars have carried out a lot of research work on negative group delay [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. An equalization technique based on negative group delay is proposed in [13,14], which can reconstruct the signals changed by time delay. Negative group delay technology can be used in antenna design to improve the performance of array antennas [15,16]. A power divider with negative group delay characteristics is designed in [17]. An original circuit theory that can identify and synthesize simple band-pass negative group delay (NGD) circuit topologies is presented in [18]. A first-order low-pass negative group delay topology is proposed in [19]. The synthesis and design of high-pass NGD network impedance is given in [20]. The passive NGD circuit using an RLC resonant circuit is studied in [26]. The resonator works at different resonant frequencies to realize the broadband NGD circuit. A novel design for a dual-band negative group delay circuit (NGDC) is proposed in [27]. A full-passband linear-phase band-pass filter (BPF) equalized with negative group delay circuits (NGDC) is proposed in [28]. An absorptive bandstop filter is proposed and synthesized with prescribed negative group delay (NGD) and negative group delay bandwidth (NBW) in [29].
The existing negative group delay circuits all use one circuit to achieve a single function. This article proposes a novel multifunctional negative group delay circuit, which implements three negative group delay functions. If the element values are in different ranges, there will be three different negative group delay circuits of band-pass, high-pass and low-pass. This circuit has good negative group delay performance and insertion loss. For the negative group delay of band-pass, high-pass and low-pass circuits, the influence of element values on the center frequency, cut-off frequency and minimum negative group delay of the negative group delay circuit is given.
This paper consists of three main parts. The second section introduces the structure of this circuit. Through equation derivation, the conditions for the band-pass, high-pass and low-pass negative group delay characteristics of the circuit are demonstrated. The third section discusses the relationship between the minimum negative group delay and the element value of the negative group delay circuit. In the fourth section, the theory, simulation and measurement are carried out. The calculation and circuit simulation are given. The final part of the paper is the conclusion.

Multifunctional Negative Group Delay Circuit
A novel multifunctional negative group delay circuit is proposed. As shown in Figure 1, this circuit can realize band-pass, high-pass and low-pass negative group delays when the capacitor, inductor and resistor take different values.

Band-Pass and High-Pass Negative Group Delay Circuits Analysis
The impedance of the negative group delay circuit is Equation (1). The characteristic impedance of the circuit is Z 0 . The transfer function of the circuit is Equation (2), and the group delay is Equation (3).
According to Equation (3), when ω = 0, C 1 , R 1 , R 2 and Z 0 satisfy the Equation (4). τ denotes the group delay value at ω = 0. At this time, τ is independent of L 1 . The magnitude of τ is used as one of the bases to distinguish the delay types of negative groups.
When τ = 0, R 1 = R 2 = Z 0 = 50 Ω, according to Equation (3); two significant frequency values are calculated, which are respectively Equations (5) and (6). The bandwidth of the negative group delay circuit is shown in Equation (7). f 0 represents the frequency corresponding to the minimum value of negative group delay in Equation (8), when the circuit presents band-pass characteristics.
The circuit shown in Figure 1 is a band-pass negative group delay circuit if Equation (9) is satisfied. It is a high-pass negative group delay circuit if Equation (10) is satisfied.

Low-Pass Negative Group Delay Circuit Analysis
If there is no capacitor in Figure 1, the impedance of the circuit is shown in Equation (11). The transmission function is shown in Equation (12). A low-pass negative group delay circuit will be obtained.
According to Equation (3), when f = 0, the NGD value and L 1 satisfy the relationship (13). The NGD depends on the value of the inductance in the circuit. The larger the inductance, the smaller the negative group delay.
The cut-off frequency f c of the low-pass NGD circuit is obtained when the group delay τ = 0. According to Equations (12) and (3), a practical frequency value is calculated as f c , and the expression such as (14). When the resistance and characteristic impedance are both fixed values, the cut-off frequency decreases as the inductance increases. When Equation (15) is satisfied, the circuit is a low-pass negative group delay circuit.

Simulation and Discussion of NGD Circuit with Different Element Values
The minimum NGD, insertion loss, center frequency, cut-off frequency and other parameters of the NGD circuit will all be affected by the element value, which will be discussed separately below.
3.1. Analyze the Band-Pass NGD Circuit 3.1.1. Influence of L 1 and C 1 on Band-Pass NGD The negative group delay τ and inductance L 1 satisfy Equation (16) when the capacitance and resistance remain unchanged. Capacitance C 1 = 1.6 pF, resistance R 1 = R 2 = Z 0 = 50 Ω. When the inductance increases, the negative group delay τ becomes smaller.
When the capacitance and resistance are constant, the NGD bandwidth and center frequency decrease with the decrease of inductance. The simulation results are shown in Figure 2 and Table 1. The frequency range selected during simulation can be used for TV and data broadcasting.  Table 1. Data of band-pass negative group delay circuit with different inductance. According to Equation (3), inductance L 1 = 470 nH, and resistance R 1 = R 2 = Z 0 = 50 Ω are substituted into Equation (3). Equation (17) is obtained. The capacitance C 1 has almost no effect on the negative group delay, but the center frequency f 0 of the minimum negative group delay varies with the capacitance C 1 . When the capacitance C 1 is changed, and the inductance L 1 = 470 nH, the resistance R 1 = R 2 = Z 0 = 50 Ω, and the negative group delay remains τ ≈ −6.26 ns.  (2) and (3), the resistance R 2 , the inductance L 1 , and the capacitor C 1 remain unchanged, and the resistance R 1 is changed. The smaller the resistance R 1 , the smaller the negative group delay value. Negative group delay τ(f 0 ) ≈ −146 ns when R 1 = 5 Ω, and it is τ(f 0 ) ≈ −250 ns when R 1 = 0.5 Ω. The simulation results are shown in Figure 3, and the performance data are shown in Table 2. The frequency of about 21 MHz can be used for long-distance short-wave communication.

Influence of R 2 on Band-Pass NGD
The resistance R 2 mainly affects the bandwidth of negative group delay circuits. When changing the resistance R 2 , the resistance R 1 , the inductance L 1 and the capacitance C 1 remain unchanged. When the resistance value R 2 increases, the bandwidth of the negative group delay circuit increases. The negative group delay basically remains unchanged when R 2 is greater than 50 Ω. The insertion loss at f 0 is about 3.5 dB which basically remains unchanged. The simulation results are shown in Figure 4, and the performance data are shown in Table 3.

High-Pass NGD Circuit
According to the values of L 1 , C 1 , R 1 and R 2 in Table 4, the circuit presents a highpass negative group delay circuit. The negative group delay decreases with the increase of capacitance, while the inductance and resistance remain unchanged. The simulation results are shown in Figure 5. The frequency range of 461-671 kHz is suitable for marine communication and medium range navigation.

Low-Pass NGD Circuit
The circuit appears as a low-pass NGD circuit according to the values of components L 1 , C 1 , R 1 and R 2 in Table 5. According to Equation (13), when the inductance decreases, the negative group delay increases. The cutoff frequency and the bandwidth increase. The simulation results are shown in Figure 6, from which the influence of inductance on the parameters of the low-pass negative group delay circuit can be seen.   (2) and (3). In the simulation, the Murata models of capacitors and inductors imported in ADS are used. These capacitor and inductor models include theoretical parameters and parasitic parameters. Due to the influence of parasitic parameters, the simulation results deviate from the theoretical calculation. The simulation results are different from the measured results in the minimum negative group delay and the center frequency corresponding to the minimum negative group delay, which is due to the influence of the actual element accuracy, PCB substrate and routing.  The comparison of the band-pass NGD with the reported literature is shown in Table 6. Compared with [18], the negative group delay is reduced by 4.26 ns, the bandwidth is increased by 8.78 MHz and the insertion loss is decreased by about 11.5 dB. Compared with paper [21], the negative group delay is reduced by 0.66 ns, and compared with paper [22], the negative group delay is reduced by 3.85 ns, and the insertion loss is decreased by 4.39 dB. It can be seen from the comparison that this design has better negative group delay and insertion loss.

High-Pass NGD Circuit
When the element values are C 1 = 33 nF, L 1 = 3.9 nH and R 1 = R 2 = Z 0 = 50 Ω, the group delay is τ = 165 ns > 0(f ≈ 0), and the frequencies are f 2 = 1118.3 MHz, f 1 = 0.18 MHz and f 0 = 0.3 MHz under the condition of τ = 0, which meets the conditions for forming a high-pass negative group delay circuit. The fabricated high-pass NGD circuit is still shown in Figure 7a, except that the element values and the band-pass NGD circuit are different. The theoretical, simulation and measurement waveforms are shown in Figure 9a,b. It can be seen that the results of simulation and measurement are basically the same. Due to the limitations of the instrument, the measured data in Figure 9a,b are a straight line below 300 kHz. There are few studies on high-pass negative group delay. In this paper, a negative group delay of −20.5 ns is realized in the low-frequency band, and the insertion loss is small. As the capacitance value increases, the negative group delay decreases, but it is limited to the test instrument and cannot render data. The performance comparison between the high-pass negative group delay circuit and the literature is shown in Table 7.

Low-Pass NGD Circuit Analysis
The element values are L 1 = 470 nH and R 1 = R 2 = Z 0 = 50 Ω in the low-pass negative group delay circuit. Group delay is τ = −3.13 ns < 0(f = 0). The cut-off frequency is f c = 9.28 MHz when τ = 0, and the conditions of forming a low-pass negative group delay circuit are satisfied. The fabricated low-pass negative group delay circuit is shown in Figure 7a; only the element values are different from those of band-pass and high-pass.
The theoretical, simulation and measurement waveforms are shown in Figure 10a,b. It can be seen by comparison that these three results are basically the same. Compared with the literature [19], although the negative group delay is larger, the insertion loss is increased by about 5.5 dB. This paper achieves a smaller negative group delay and insertion loss compared to [22]. The insertion loss at the minimum negative group delay is similar, but this paper achieves a smaller negative group delay, which is about 3 ns smaller than [24], as shown in Table 8. Considered comprehensively, the low-pass NGD circuit originally designed has good characteristics.

Conclusions
The proposed multifunctional novel NGD circuit realizes the functions of band-pass, high-pass and low-pass negative group delay when the elements take different values, respectively. The circuit has good negative group delay characteristics, and the measured results are basically consistent with the theoretical design. Through calculation and simulation, the relationship between the elements and the negative delay circuit parameters is obtained. According to the authors' understanding, the existing NGD circuits all use one circuit to realize a single function. In this paper, three different NGD functions are realized with one circuit. These results provide a good theoretical basis for the flexible design and application of NGD circuits. In practical applications, the value of elements can be adjusted according to requirements, and various required negative group delay circuits can be obtained. This circuit is convenient to adjust, save costs and meet different application occasions. In the future, the NGD circuit can be used to realize the negative group delay of band-stop filters in the passband. and 2020-MS-127), the Liaoning Revitalization Talents Program and the Fundamental Research Funds