Issue 
Eur. Phys. J. Appl. Phys.
Volume 91, Number 1, July 2020



Article Number  10903  
Number of page(s)  11  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2020200016  
Published online  30 July 2020 
https://doi.org/10.1051/epjap/2020200016
Regular Article
Negative group delay experimentation with tee connector and cable structures
^{1}
Nanjing University of Information Science & Technology, Nanjing, China
^{2}
Amity University Haryana, Gurgaon, 122413 India
^{3}
Université Clermont Auvergne, Institut Pascal, SIGMA Clermont, 4 av Blaise Pascal, 63178 Aubière, France
^{*} email: blaise.ravelo@yahoo.fr
Received:
23
January
2020
Received in final form:
11
May
2020
Accepted:
8
June
2020
Published online: 30 July 2020
The unfamiliar negative groupdelay (NGD) phenomenon is experimented with structure constituted by teeconnectors and coaxial cables. The topology of the NGD generator is described. It consists of two parallel 50Ω characteristic impedance cables. The Sparameter model in function of the cable physical lengths is established. The existence condition enabling to understand the NGD phenomenon is defined. A proof of concept simply is constituted by two threeport SMA connectors, three SMA transitions and 13cm length cables. The NGD experimentation is performed from 1.5 GHz to 3.5 GHz. Triband NGD aspects in good agreement with the theory and simulation is observed experimentally. Particularly, high figureofmerit NGD circuit with −3 ns NGD level, less than 2 dB insertion loss and 12 dB return loss around 3 GHz centre frequency is measured.
© EDP Sciences, 2020
1 Introduction
The group delay (GD) may affect the transmission signal quality in the communication systems [1–6]. In addition, with the dispersion effect [1], the group delay can become non flat in the operation frequency bands of RF and microwave components. Depending on the application area, such degradation effects are interpreted with different analytical parameters group [1,2], propagation [3,4] and time [5,6] delays. Therefore, an efficient technique is necessary for estimating the influence of delays on the communication quality [5,6]. In addition, effective technical solutions have been deployed to overcome the delay effects as the development of design methods dedicated to group delay equalizers [7–9], minimizer [10] and linearizer [11] were proposed. So far, different techniques to design equalizers were introduced with typically allpass network filters [12]. Nevertheless, these types of equalizers have negative undesirable effects as the generation of additional positive group delays. The uncontrollable group delays are not good and may degrade the system performance.
An alternative technique of equalizer implementation using negative group delay (NGD) function was introduced in [13]. This type of NGD equalizer enables to realize outstanding solutions as the neutralization of undesirable RC and LCeffects generally degrading the signals propagating through the electrical interconnections [4]. However, nowadays, in difference to the other classical functions, the NGD phenomenon is still not understandable for most of electronics, RF and microwave engineers. For this reason, the present paper investigates an experimentation of NGD phenomenon with a noncomplicated electrical structure.
Since the first experimentation of NGD phenomenon in 1990s, few groups of research are currently working on this counterintuitive topic. Nevertheless, various non mature attempts of applications in particular in the area of RF and microwave engineering were proposed [13–26]. As reported in [14], an NGD compensation method of oscillators, filters and communication systems was initiated [14]. Another compensation method of array antenna effect based on the NGD circuit fabricated with lossless double negative metamaterials can be found in [15]. Design of matched bridged tee network with positive phase slope was studied in [16]. But, the passive NGD circuits used in these applicative proposals suffer from attenuation losses. To alleviate this roadblock, alternative applications with NGD active circuits were proposed [17–22]. For example, a design method of enhanced bandwidth feedforward amplifier using NGD resonator was presented in [17,18]. More recently, another design of bidirectional amplifier with NGD matching circuits was investigated in [19–21]. A design method of distributed amplifier using reconfigurable NGD circuit is presented in [22]. Different antenna design applications were also proposed as the arbitraryangle squintfree beamforming in seriesfed antenna arrays using nonFoster NGD networks [23]. Then, design method of arbitrary terminated unequal coupler with bandwidthenhancement by using NGD circuit is proposed in [24]. Designs of NGD devices and circuits for RF and microwave front and backend chains based were also invented [25,26]. In spite of those applications, today, the NGD function can be classified as the most uncommon electronic function for most of researchers and engineers.
To feed our knowledge on the uncommon NGD phenomenon, a brief stateoftheart is narrated in this paragraph. Basically, the NGD phenomenon can be identified with its outstanding signature of the apparition of output signal in advance of smoothed input [27]. Knowing the initial interpretation of the group delay [28,29], it can be understood that the NGD phenomenon can be generated systematically when a linear circuit transmission phase presents a positive slope when the frequency increases [28,29]. The first NGD synthesizer with typically RLCresonator passive network was investigated in [30,31]. One decade later, NGD passive resonant circuits inspired from double negative metamaterial structures were designed [32–34]. The NGD function meaning in timedomain with the negative delay was theoretically investigated and experimented in [27]. However, it was emphasized in [35] that the NGD circuit must operate with limited timeadvance. Nevertheless, last decade, some innovative circuits operate with NGD phenomenon were developed [36–46]. NonFoster reactive elements NGD passive networks were proposed [36]. Maximally flat NGD active circuit by exploiting the transversal filter concept is proposed [37]. Then, design of passive NGD lumped circuit with modified lossy lefthanded metamaterial structure is proposed in [38]. Mostly, the NGD circuits using lumped passive elements are significantly lossy. Therefore, NGD circuits with distributed transmission lines (TLs) were proposed [39–45]. Transmission type NGD distributed passive circuits were also introduced [39–41]. Another NGD distributed passive circuits based on the coupling effect were proposed in [44,45]. This tremendous progress of NGD design should be accompanied to pedagogical experimental open technique to make the topic more familiar to nonspecialists.
As discussed previously, further illustrative theory and experimentation is necessary to illustrate and explain the NGD phenomenon. More research work must be performed to open the NGD topic to nonspecialist including RF and microwave engineering community. Simpler NGD topology operating with lower loss and good access matching remains an open topic of research. For this reason, the present paper introduces an easy to understand NGD investigation. To do this, particularly simple structure constituted by SMA TeeConnectors and SMA cables operating as a bandpass NGD function is presented. In difference to the previous NGD study [13–27,30–45], the present work develops an innovative theory and experimentation demonstrating the possibility to generate the NGD phenomenon in function of the SMA cable length and attenuation. Thus, the proposed structure is helpful for understanding the NGD function. It can be experimented easily with simple SMA connectors and cables by nonspecialists.
The paper is organized in four main sections. Section 2 describes the theoretical investigation of the TeeCable topology. The proposed topology is designed with only distributed elements without lossy lumped elements. The circuit theory is elaborated from the Sparameter modelling. The NGD analyses will be introduced in Section 3. The NGD existence equations and centre frequencies in function of the cable delay and attenuation are investigated. Section 4 is focused on the practical validations including the NGD experimentation of the tee connector combined with coaxial cables. Proofofconcept (POC) built with TeeCable based structure will be described. Comparisons between the analytically calculated, simulated and experimented results will be discussed. Then, the last Section presents the conclusion of the paper.
2 NGD general theory on tee connector and cable structure
This section is focused on the NGD theory. The equivalent circuit of the TeeCable structure will be introduced. The Sparameter model will be calculated. The group delay expression will be established in order to perform the NGD analysis.
2.1 Description of the tee connector and cable structure
The NGD microwave circuit under study behaves as a twoport passive topology. In difference to work published in [13–27], the present NGD topology is constituted by interconnected threeport Tee connectors and cables represented by TLs. The connector terminals are referenced by the elements with connection Ports ①–②–③ and Ports ④–⑤–⑥. For the sake of mathematical simplicity, the constituting connectors are represented by ideal three port Smatrix. As depicted in Figure 1, connector port ① and port ⑥ represent respectively as the overall NGD topology main input and output.
The overall circuit is loaded by R_{0} = 50 Ω which is assumed as the reference impedance. Ports ② and ④ are directly connected and ports ③ and ⑤ are interconnected through a lossy cable with characteristic impedance R_{0} and physical length d. The variables a_{m} and b_{m} (m = {1,…,6}) represent the input and output wave powers propagating through the structure branches. The proposed topology Sparameter modelling will be described in the next paragraph.
Fig. 1 Equivalent schematic of the combined tee connectors and cables network under study. 
2.2 Theoretical modelling of Smatrix
This analytical study is elaborated by supposing that the Tee connector with negligible physical length and the cable is perfectly matched. By hypothesis, the threeport Tee connector Sparameter model is symmetrically expressed as:(1)
This Tee connector Smatrix can be operated with the topological parameters of the circuit based on the following matrix relation:(2)and:(3)
In this paper, the interconnect lossy cable characteristic impedance is assumed equal to Z_{c} = R_{0}. By denoting v the wave speed, the cable physical length is expressed as:(4)
The lossy cable is characterized by the parameter:(5)
with α is the attenuation constant. Under this hypothesis, by denoting the angular frequency variable ω, the interconnect reflection and insertion losses introduced in [39–45] become:(6) (7)with:(8)and(9)are the cable delay. These Sparameters are linked to the access port wave powers by the matrix relation:(10) (11)
Consequently, we have the power wave expressions:(12) (13)
2.3 Frequency dependent expression of Sparameter coefficients
According to the Sparameter theory, the considered twoport topology main access wave powers are linked by the relation:(14)
These coefficients can be expressed in function of the circuit parameters by calculating the output wave powers b_{1} and b_{6}, and the input wave power a_{1}. First, b_{1} and b_{6} can be determined by successive combination of the wave powers propagating through the cable and connector access ports. The analytical expressions obtained from the Smatrices defined previously in equations (1), (6) and (7) give the simplified formulas: (15) (16)
The associated reflection coefficient magnitude S_{11}(ω) = S_{11}(jω) is written as: (17)
The transmission coefficient magnitude S_{21}(ω) =S_{21}(jω) is given by: (18)
With the presence of sin(.) and cos(.) terms, we can understand that these magnitudes are periodical function.
2.4 Frequency dependent expression of GD
By denoting the transmission phase associated to S_{21} is defined as:(19)
In circuit theory, the NGD phenomenon can be easily understood based on the group delay (GD) analytical definition. To do this, it is worth to recall this definition. The analytical expression of GD was initiated in [26,27]:(20)
The proposed cell NGD fundamental properties and characteristics will be deduced from this transmission coefficient expression. The associated transmission phase is:(21)with:(22)and:(23)where:(24) (25)
It yields the GD defined earlier in (20) can be formulated as:(26)with:(27) (28)
It can be pointed out that the frequency responses (transmission and reflection coefficients, and the group delay behave as periodical functions depending on the parameters τ_{0} and Δτ. Therefore, we are proposing to conduct the analytical analyses at some particular frequencies in the next paragraph.
3 NGD analysis at particular frequencies
This section develops the main physical approach on the NGD analysis.
3.1 Identification of particular frequencies
For the preliminary analytical observation, the NGD analysis can be performed at the particular values of angular frequencies. The reference angular frequency is linked to the quarter wavelength of the cable physical length:(30)
The analyses are performed at very low frequencies ω ≈ 0 and the multiple of the particular angular frequency ω_{0}:(31)with the integer m = {0,1,2,…}. It can be speculated the behavior of the teecable network at the angular frequencies:(32) (33) (34)
3.2 Insertion and reflection losses and group delay at very low frequencies ω ≈ 0
The reflection and insertion losses expressed in (19) and (20) are transformed as respectively:(35) (36)
It can be understood from formula (35) that the input and output reflection can be matched. The group delay expressed in (26) becomes:(37)
This formula enables to predict the influence of the parameters τ_{0} and Δτ. So, the group delay of the Teecable topology is unconditionally positive at very low frequencies ω ≈ 0. Therefore, the circuit cannot operate as a lowpass NGD function. Therefore, let us see in the next paragraph the NGD values at the frequencies expressed in (31), (32), (33) and (34) if there is possibility of bandpass NGD function.
To understand more generally the teecable behavior, we are proposing to calculate the reflection and transmission coefficients, and the group delay in function of:(38)or(39)
3.3 Insertion and reflection losses at particular frequencies ω(m)
At the particular frequencies ω(m), the reflection and insertion losses introduced in (17) and (18) are transformed as respectively: (40) (41)
However, these general expressions of the reflection and insertion losses, and the group delay in function of m are rather sophisticated. Therefore, we can proceed with the Maclaurin expanding with respect to x for example limited to third order. Consequently, reflection and insertion losses introduced in formulas (40) and (41) are reduced as respectively:(44) (45)
3.4 GD expressions at ω = ω(m)
It is established that at the particular frequencies ω(m), the group delay expressed in (29) becomes: (46)
This NGD is mathematically negative under the following condition: (47)
Knowing that a < 1, condition (47) is impossible to achieve because the quantity:(49)is always positive. The existence of condition (47) depends on the following discriminant of the group delay quantity numerator with respect to the variable x:(50)
It can be discriminated that:(51)the group delay expressed in (46) is always negative independently to the variable x,(52)
the group delay can be negative when x fulfils the condition:(53)
Table 1 presents the group delays at the frequencies expressed in (31), (32), (33) and (34) in function of the parameters a, τ_{0} and Δτ. As predicted by equation (46), the NGD level absolute values increases with m. As illustrated in formulas (54), (55) and (56), the teecable topology may behave as a bandpass NGD function.
As concrete verification, validation results are discussed in the next section. The POC structure will be described. Comparisons between the modelled, simulated and measured results will be examined.
Group delays at particular angular frequencies.
4 Simulation and experimental demonstrations
To verify the effectiveness of the previous theory, validation results are discussed in the present section. First, numerical feasibility analyses in function of the cable parameters is elaborated based on ideal circuit parametric sweeping. The POC of the tee connector combined with cables is described. Then, the simulated and experimental results are compared with the calculated modelled results.
4.1 Parametric analyses with ideal structures
To get further insight about the influence of the cable length on the NGD circuit, parametric analyses are conducted. Figure 2 depicts the schematic of the simulated structure. The teecable network was simulated in the SPICE schematic environment of the ADS^{®} simulator from Keysight Technologies^{®}. It is composed of ideal coaxial tees and cables. The two identical connectors are defined by the characteristic impedance R_{0} = 50 Ω, physical length d_{c} = 4 mm and relative permittivity ε_{r} = 2. The two cables present the attenuation a = −0.3 dB and different lengths d and d + Δd.
The present parametric analyses aim to illustrate the influence of the delays τ_{0} and Δτ. To gain more practical approach, the cable lengths are defined by the arbitrary chosen values d = 0.12 m and Δd varied from 3.5 mm to 5.5 mm step 0.5 mm, and d varied from 0.1 m to 0.15 m step 0.02 m with Δd = 4.5 mm. The structure Sparameters were simulated from 1.5 GHz to 3.5 GHz. The group delay and Sparameter frequency responses illustrating the influence of Δd are displayed in Figures 3 and 4 respectively. The NGD center frequency decreases and the group delay level increases with the cable length difference Δd. It is seen that the teecable network enables to achieve a multiband NGD behaviors in the GHz frequency band. In addition, an outstanding NGD level better than −1 ns over the insertion loss better than 2 dB and reflection loss better than −15 dB in the NGD bandwidth. The group delay and Sparameter frequency responses illustrating the influence of d are displayed in Figure 5.
The NGD central frequency changes significantly with d as shown in Figure 5a. A frequency shift of the NGD multiband effect is observed. NGD level varying from about −1 ns to −6 ns in function of the central frequency position. Furthermore, as illustrated by Figures 5b and c, the insertion and return losses, respectively do not present notable change with length d.
Fig. 2 Schematic of the simulated teecable network. 
Fig. 3 Group delay response of the teecable network with the influence of Δd. 
Fig. 4 Transmission and reflection coefficients of the teecable network with the influence of Δd. 
Fig. 5 Group delay, transmission and reflection coefficients of the teecable network with the influence of d. 
4.2 Description of teecable POC
The photograph of the simple 50 Ω coaxial teecable POC prototype is exposed in Figure 6a. The tee and cable are classical instruments used in microwave measurement engineering implemented in SMA technology. The operated cable lengths are d ≈ 0.125 m and Δd ≈ 3.5 mm. They are built with concentric dielectric with relative permittivity ε_{r} = 2.3 and loss tangent tan(δ) ≈ 0.01. The inner and outer radius of the dielectric insulating are about 3.44 mm and 6.76 mm. The constituting conductor element is a copper having thickness t = 0.5 mm. The Sparameter measurements are completed using a VNA from Rohde & Schwarz (ZNB 20, frequency band 100 kHz to 20 GHz) which is shown in Figure 6b. The repeatability of this Sparameter experimentation is guaranteed like the measurement of classical microstrip circuits.
Fig. 6 Photographs of (a) the tested teecable prototype and (b) the measurement experimental setup. 
4.3 Discussion on simulated and measured results
The Sparameters of the tested teecable network were measured in the frequency band from 1.5 GHz to 3.5 GHz step 10 MHz. Figure 7 show the comparisons between the theoretical model, simulation and measurement of the group delays. The particular frequency calculated from (30) is f_{0} ≈ 0.79 GHz. The modelled, simulated and measured results are in good agreement. Original triband NGD behaviour is observed at around 2f_{0} ≈ 1.58 GHz, 3f_{0} ≈ 2.42 GHz and 4f_{0} ≈ 3.19 GHz. As seen in Figure 7, the NGD presenting optimal level of about −1 ns, −2 ns and −3 ns were observed around these frequencies respectively. However, the NGD bandwidths are particularly narrow band less than 12 MHz.
The measured NGD circuit enables to achieve insertion losses better than 1 dB and reflection loss better than 15 dB at the first NGD centre frequency as illustrated in Figure 8. The obtained insertion loss is widely better compared to the existing NGD passive circuits. These theoretical, simulation and experimental results confirm the NGD concept feasibility with the teecable networkbased topology. The discrepancies between the modelled, simulated and measured Sparameters can be caused by the fabrication imperfections, the tee and cable constituting material characteristic dispersions, the cable attenuation, which is assumed theoretically constant, changes with the frequency in function of the conductivity, and numerical inaccuracies of the simulator.
The observed discrepancies between the calculated, simulated and measured results shown in Figure 7 and in Figure 8 are essentially caused by the following imperfections:

Dispersions on the parameters of the cable substrate material,

Mismatch induced by the Tee cables/connectors (see Fig. 6),

Numerical computation drifts of the full wave simulating tool, and of the calculated tool; this includes for instance the dispersion models and simulations constitutive parameters (geometrical, electrical ones).
Fig. 7 Comparison of the modeled, simulated and measured group delay of the teecable NGD POC prototypes shown in Figure 6. 
Fig. 8 Comparison of the modeled, simulated and measured reflection and transmission coefficients of the teecable NGD POC prototype shown in Figure 6. 
4.4 Comparison of NGD specifications with the stateoftheart
We can understand from the comparative values of Table 2 the basic performances of the parallel cable structure topology under investigation and the existing ones available in the literature (see [37,46–51]). It can be underlined that the parallel cable NGD structure enable to achieve:

low signal attenuation in the NGD bandwidth,

reflection loss better than −14 dB in the NGD bandwidth,

and it can be implemented with fully distributed elements without lossy lumped component.
However, the parallel cable drawbacks are:

the large size of the cable which is not easy to integrate in the printed circuit boards,

and the imperfections of the cable fabrications may affect undesirably the bandpass NGD function parameters.
Comparison of the NGD structure specifications and the available ones in the stateoftheart.
5 Conclusion
An NGD analytical and experimental study on outstanding simple teecable network is investigated. The Smatrix model of the teecable equivalent circuit is developed. The NGD analysis is introduced in function of the cable parameter. The NGD theory validation is confirmed by calculations, simulations and experimentations. Parametric and numerical analyses of Sparameters in function of the cable length and attenuation are discussed. In addition, as POC, a teecable network prototype is tested. The theoretical prediction is verified with both simulations and measurements. It is demonstrated that the tested tee cable generates a multiband NGD effect between 1.5 and 3.5 GHz. In difference to the existing NGD passive circuits, the teecable NGD topology allows to achieve very low insertion loss, better than 2 dB and reflection loss better than 12 dB.
Author contribution statement
Fayu Wan performed the main writing of the paper and was a part of NUIST, China. Xiaoyu Huang performed the test and the data processing and was a Master student of NUIST, China. Preeti Thakur and Atul Thakur contributed on the technical and English correction of the paper. Sébastien Lalléchère performed result verifications by simulation. Blaise Ravelo was the main initiator of the NGD cableconnector structure and the development of the NGD theory. All authors contributed on the article writing with critical reviews and corrections.
Acknowledgments
This research work was supported in part by NSFC under Grant 61601233 and Grant 61750110535, in part by the Defense Research Foundation under Grant 6140209050116ZK53001, in part by NSF of Jiangsu under Grant BK20150918, in part by the Jiangsu Innovation and Enterprise Group Talents Plan 2015 under Grant SRCB201526, and in part by PAPD.
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Cite this article as: Fayu Wan, Xiaoyu Huangi, Preeti Thakur, Atul Thakur, Sébastien Lalléchère, Blaise Ravelo, Negative group delay experimentation with tee connector and cable structures, Eur. Phys. J. Appl. Phys. 91, 10903 (2020)
All Tables
Comparison of the NGD structure specifications and the available ones in the stateoftheart.
All Figures
Fig. 1 Equivalent schematic of the combined tee connectors and cables network under study. 

In the text 
Fig. 2 Schematic of the simulated teecable network. 

In the text 
Fig. 3 Group delay response of the teecable network with the influence of Δd. 

In the text 
Fig. 4 Transmission and reflection coefficients of the teecable network with the influence of Δd. 

In the text 
Fig. 5 Group delay, transmission and reflection coefficients of the teecable network with the influence of d. 

In the text 
Fig. 6 Photographs of (a) the tested teecable prototype and (b) the measurement experimental setup. 

In the text 
Fig. 7 Comparison of the modeled, simulated and measured group delay of the teecable NGD POC prototypes shown in Figure 6. 

In the text 
Fig. 8 Comparison of the modeled, simulated and measured reflection and transmission coefficients of the teecable NGD POC prototype shown in Figure 6. 

In the text 
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