0IO-Shape PCB Trace Negative Group-Delay Analysis

This paper elaborates a negative group delay (NGD) analysis of 0IO-shape printed circuit board (PCB) traces. This circuit topology is originally implemented with a tri-coupled line (3CL) six-port element with the lateral side connected through lossy transmission lines (TLs). After description of the electrical equivalent diagram, the S-matrix model is established. The group delay (GD) is formulated from the transmission coefficient as a function of the 0IO topological parameters. The effectiveness of the GD modelling is verified with a microstrip circuit proof-of-concept (POC). Simulations and measurements, which are in good agreement, confirm the dual-band bandpass NGD behavior of the 0IO POC. The fabricated prototype generates NGD levels better than −1 ns at NGD center frequencies of about 2.2 GHz and 3 GHz. In addition, to this good NGD performance, the 0IO POC operates with a low insertion loss better than 2.5 dB and reflection losses better than 12 dB in the NGD bandwidths.


I. INTRODUCTION
The group delay (GD) is a key parameter in microwave electronic circuit systems. The GD was exploited for design electronic functions as phase shifting [1], antenna arrays [2] and feedforward amplifier [3] etc. To deal with the unexpected GD issue, the extra delay line is regularly used. The compensation consists generally in compensating the unbalanced GD in the RF and microwave components as in feedforward amplifier [4]. However, this classical solution may be penalizing by adding delay line. Furthermore, the delay line structure can also increase the circuit size greatly. To overcome this issue, recently, negative group delay (NGD) circuits are thought to be a good solution to equalize the microwave electronic system delay.
The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso .
In [1], the NGD function was applied to phase shifter to generate at and constant phases over a broad bandwidth. A transmission lines loaded active NGD network is introduced in [2] to solve the beam-squinting problem of conventional series-fed antenna arrays. An NGD circuit is used to enhance feedforward amplifier efficiency by eliminating the delay element, which is one of the major sources of efficiency degradation, without affecting the linearization performance [3]. The NGD effect was also used to improve the flatness of power divider [5]. The NGD structure is constructed with a two-way microstrip line power divider with equal power division ratio. The power divider different transmission paths generate NGD effect despite the insertion losses. From above, it can be seen that the NGD circuits have very extensive applications on RF/microwave devices and systems.
The first NGD synthesizers, with microwave passive circuits were proposed in [6], [7] and with very low-frequency VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ (LF) active circuits were introduced in [8], [9]. The NGD passive circuits using lumped R, L and C elements only work at frequencies lower than 1 GHz [10], [11], limiting the available scope of microwave applications. Since NGD occurs at the range of frequencies where the absorption or the attenuation is maximum, the NGD generation is accompanied by excessive isolation loss more than 20 dB [6], [7]. Therefore, the cascading of active elements such as amplifiers is a practical compensation for these attenuations [12], [13]. Thus, in [13], a field effect transistor (FET) is applied to design an active NGD circuit with 2dB gain. However, it was found that these active NGD circuits would unavoidably suffer from design inflexibility restrictions about the fixed component values and increasing design difficulties in the microwave band. And also, it can increase the out-of-band noise as well as make the circuit more complicated. Moreover, such an active topology is rather complex to design and difficult to integrate because of the lossy lumped inductor. As a result, simpler and low-loss passive topologies built with distributed transmission lines (TLs) were implemented [12]- [19]. To reduce the attenuation lower than 10 dB, a coupled line based NGD circuit is designed in [16] whose attenuation is decreased to 7.43 dB. A parallel interconnect line (PIL) NGD circuit is designed in [18]. This PIL NGD circuit is able to operate with attenuation of about 5 dB. An NGD circuit consists of the isolated-and coupledaccesses connected in a feedback loop as a coupling between ''1'' and ''0'' shape interconnect line which presents 2.4 dB insertion loss [19]. Such geometrical shape as ''01O'' sensitively with electromagnetic interference (EMI) can be found in the PCB traces.
These ''01O'' configurations can be found in PCB layouts constituted by copper plane hotspots, vias interconnects, serpentine and twirling paths (for length compensation, pads interconnects, symmetrical breakouts etc.). It would be important to investigate the coupling effect modelling of this structure and the possibility of NGD effect generation. The motivation of this work is to properly assess the impact of NGD generation (quantifying and/or evaluating it) with respect to tri-coupled line EMC (electromagnetic compatibility) and EMI scenarios. When dealing with coupling scenarios involving ''01O'' structures, it is of utmost importance for the EMC designer to ensure the system functioning taking into account NGD effects. Compared with the existing singleband NGD circuits [12]- [19], [21]- [23], the design of dualband NGD circuits [24]- [27] remains a challenging task. In order to enable the NGD circuit to operate in different frequency bands, few studies are focused on the design of dual-band NGD circuits. A dual-plane U-shaped defected structure is used to realize a dual band NGD circuit [25], whose first center frequency and second center frequency are determined by a defected microstrip structure (DMS) and a defected ground structure (DGS), respectively. However, the attenuation is worse than 44 dB. A compact dual-band NGD circuit composed of an open-circuited TL and two resistors connected by two TLs is proposed in [27]. The attenuation  is worse than 16 dB. Therefore, it is especially important to design a low-loss dual-band NGD circuit.
For this reason, in this paper, a topology of dual band NGD passive topology based on fully distributed TLs with further challenge on TL loss and delay effect is developed. The challenging NGD topology presents an innovative shape similar to ''0IO'' shape geometry. The topology is built with a tri-coupled line (3CL) six-port element with the lateral side connected through lossy transmission lines (TLs). The paper is mainly organized in four different sections as follows: • The theoretical approach allowing to model the 0IO topology S-matrix is described in Section II. After the GD modelling, the NGD analysis is introduced.
• The validity of the bandpass NGD functioning of the 0IO topology is validated in Section III via excellent comparisons between calculation, simulation and measurement.
• Finally, Section IV concludes the paper.

II. NGD THEORETICAL INVESTIGATION ON ''0lO'' TOPOLOGY
The present section is focused on the NGD theoretical approach of the 0IO topology. The S-matrix model is established from wave power interactions between the topology constituting elements. The GD expression is derived from the transmission coefficient. Then, NGD analysis is introduced in function of the 0IO parameters.
A. DESCRIPTION OF THE 0IO TOPOLOGY Fig. 2 sketches the topology of 0IO structure under study. It is comprised mainly of a six-port tri-coupled line (3CL) denoted 97708 VOLUME 8, 2020 CL(R 0 , k) having characteristic impedance R 0 = 50 and the adjacent line coupling coefficient k. We suppose that the coupling between line TL z−{ and TL |−} are negligible. The main access ports are constituted by x and y.
The lateral lines are interconnected through different lossy TLs TL k (Z k , a k , τ k ) for k = {1, 2} which are supposed to be ideal with same characteristic impedances Z k = R 0 and attenuation a k = a. However, they present different propagation delay τ 1 < τ 2 .
After the equivalent circuit introduction, the S-matrix modeling of the 0IO topology will be explored in the next paragraph. The modelling of our hexapole circuit is built with the consideration of equivalent S-parameters. The main constituting elements are a CL and the four pieces of terminal TLs introduced in Fig. 2. Based on the S-parameter theory, the general S-parameter of the 01O topology can be written in function of the wave powers. The CL six-port S-matrix constituting the central element is linked to the input and output wave powers a m and b m (m = {1, 2, . . . , 6}) by the relation: (1) The main objective of the theoretical approach is to determine the overall equivalent two-dimension S-matrix [S] 0IO by reducing this six-dimension matrix. Meanwhile, this total S-matrix can be determined from the constituting TL and CL ones. Substituting the expressions of wave powers a m and b m (m = {3, . . . , 6}) into (1), the overall S-matrix is defined by the relation: (2)

1) S-MATRIX OF CL ELEMENT
According to the coupler theory, the CL S-matrix can be written in function of the two-neighboring lines coupling (between ports x-z, x-|, y-{ and y-}) k and direct transmission (between ports x-y , z-{ and |-}): with the complex number j = √ −1. Under this hypothesis, the CL S-matrix model is ideally assigned as: 2) S-MATRIX MODEL OF TL1,2 Following the topological description introduced in Fig. 3, the TL k for k = {1, 2} S-matrices are linked to the wave powers by the relation: By denoting ω is the angular frequency variable, let us take: Therefore, the equivalent TL 1 and TL 2 S-matrices can be written as: 3

) EXPRESSION OF 0IO STRUCTURE GLOBAL S-MATRIX
It can be derived from (4) and (6) (4), the 0IO S-matrix introduced in (2) is written as: with the transmission coefficient: Based on this last expression, the NGD analysis is introduced in the next subsection.

C. ANALYSES OF S-MATRIX FREQUENCY RESPONSES
The transmission coefficient magnitude, S 21 (ω) = |S 21 (jω)|, can be formulated as: The transmission phase is defined by:

D. NGD ANALYSIS OF 0IO STRUCTURE
The 0IO topology GD is defined from the transmission phase expressed in (11) with the equation: Thanks to equations (12) and (13), we can rewrite this GD as follows: with: and In details, the phase numerator GD is given by: with: The phase denominator GD is expressed as: with: Based on these analytical expressions, parametric analyses versus TL delays τ 1,2 where performed to visualize the NGD effect. More importantly, simulations and experimental studies with a POC were also realized. The obtained validation results will be explored in the next section.

III. EXPERIMENTAL VALIDATIONS WITH 0IO PROTOTYPES
As application of the previous theory, a prototype of 0IO POC has been designed, simulated and fabricated. The obtained results will be discussed in the next paragraph.

A. DESCRIPTION OF CONSIDERED 0IO NGD PROTOTYPES
To verify the relevance of the previous theory, 0IO POCs were designed and simulated with ADS R . The designed and photographed two-port circuits based on: • FR4-epoxy substrate are viewed in Fig. 3(a) and in Fig. 3(c), • and based on Rogers substrate are shown in Fig. 3(b) and in Fig. 3(d), respectively. The designed circuits were simulated in the ADS R environment. We can see in these designs the lateral delay lines  Fig. 3(b) and Fig. 3(d). and the middle CL. Additional access lines TL 3 and TL 4 are added in order to facilitate the measurement configuration. The pictures of the 0IO POC prototype are shown in Fig. 3(c) and Fig. 3(d). These prototypes are implemented in microstrip technology without use of lumped and lossy component. The physical parameters of the FR4 and Rogers substrate 0IO circuit prototypes are indicated in Tables 1 and 2, respectively. The conductor lines are Cu-metallized. The physical sizes of constituting distributed lines TL 1 , TL 2 and CL are also indicated in Table 1. The associated electrical parameters as characteristic impedances, delay and coupling coefficient calculated from ADS R LineCalc microwave circuit calculation tool are also given in this table.
Based on this prototype, validation studies were performed with parametric analyses with respect to the TL lengths. Furthermore, practical investigation with experimental testing will be discussed in the following subsection.

B. PARAMETRIC ANALYSES
To get a predictive insight about the bandpass NGD behavior of the 0IO topology, parametric analyses with respect to TL 1 and TL 2 delays τ 1 and τ 2 , by means of physical lengths d 1 and d 2 were computed with S-parameter simulations from 2 GHz to 3.4 GHz. The obtained results are explored in the following paragraphs.

1) INFLUENCE OF TL 1 DELAY
The influence of TL 1 delay is studied with parametric simulations by varying physical length d 1 from 35 mm to 45 mm. Fig. 4(a) exposes the GD map versus frequency and d 1 . This result reveals that the 0IO topology behaves as a dual-band bandpass NGD function. Two NGD bandwidths (BWs) can be identified. The first NGD BW situated around 2.2 GHz is zoomed in Fig. 4(b).
It shows an NGD function that is not sensitive to d 1 variation. It means that this NGD bandwidth should depend to the loop related to TL 2 and therefore, it must be linked to delay τ 2 . The second NGD BW area, situated between 2.73 GHz and 3.33 GHz is displayed in Fig. 4(c). This NGD BW varies inversely with delay τ 1 . Therefore, we denote f 1 the NGD center frequency associated to this second NGD bandwidth.
The analytical relations between the two NGD center frequencies f 1,2 and delays τ 1 can be understood more clearly with the graphical plots of Fig. 5(a). At these NGD center frequencies, NGDs are approximately estimated as GD(f 1 ) varying between −1.3 ns and −1.2 ns, and GD(f 2 ) ≈ −1.4 ns. In addition to the GD analysis, the S 21 variation with respect to d 1 is displayed in Figs. 6. The low loss aspect with insertion VOLUME 8, 2020  loss lower than 2.5 dB is explained by the wide frequency range mapping in Fig. 6(a). It can be underlined that the insertion loss presents a similar behavior as the GD shown in Figs. 4. The insertion loss in the first NGD bandwidth is insensitive to the variation of d 1 .

2) INFLUENCE OF TL 2 DELAY
Similar to the previous case of study, parametric analyses with respect to d 2 varied from 55 mm to 65 mm were realized to investigate the influence of TL 2 delay τ 2 . Fig. 7(a) displays the maps of GD from 2 GHz to 3.4 GHz. This map reveals that the first NGD BW with center frequency f 2 is sensitive to delay τ 2 .  As mapped in Fig. 7(b) and plotted in Fig. 8(a), the NGD center frequencies f 1 ≈ 3 GHz and f 2 varies between 2 GHz and 2.4 GHz. As seen in Fig. 8(b), the associated GDs are approximately equal to τ 1 ≈ −1.25 ns and τ 2 ≈ −1.4 ns. Figs. 9 display the maps of transmission coefficient versus frequency and τ 2 . Once again, the map behaviors are similar to the GD and shows that only the first NGD BW is sensitive to τ 2 .

C. INVESTIGATION ON SIMULATED AND EXPERIMENTAL RESULTS
To complete the validation, practical analysis of 0IO NGD prototype was investigated via S-parameter measurement  Fig. 3(a) and Fig. 3(c).  from 2 GHz to 3.4 GHz. The experimental setup configuration with vector network analyzer (VNA) is shown in Fig. 10.

Two different ''01O'' interconnect prototypes from FR4-and
Rogers-substrate based were tested to illustrate the NGD effect possibility.

1) DISCUSSION ON NGD TEST RESULTS OF FR4-BASED 0IO PROTOTYPE
Figs. 11 present the comparison results in the wide frequency range which confirm the dual-band bandpass NGD behavior of FR4 substrate based 0IO prototype. The simulated and measured GD and transmission coefficients of Figs. 11(a) and 11(b), respectively are in very good agreement. However, as shown in Fig.11(c), the simulated and measured reflection coefficients are showing considerable differences of behaviors. These main differences are first of all linked to the low values of reflection coefficient under than −10 dB where the differences seem to be relatively significant but remain negligible in linear valuer. VOLUME 8, 2020 Fig. 3(b) and Fig. 3(d). In the all parameters (the GD, transmission and reflection coefficients), the observed discrepancies between simulated and measured results plotted in Figs. 11 are mainly due to: • The fabrication errors related to the microstrip line width, lengths and connectors, • The measurement uncertainties, • The full wave simulation inaccuracies linked to the meshing and iterative computations, • And also the FR4-epoxy substrate material parameter uncertainties as the relative permittivity, and loss tangent. As indicated in comparative Table 3, the NGD center frequencies are localized around 2.2 GHz and 3 GHz. To highlight the NGD behaviors, the zoom in plot around the NGD narrow BWs are depicted in Figs. 12. It can be seen that the FR4 substrate based 0IO prototype presents an NGD BWs of about 35 MHz and 53 MHz. The slight shifts and differences between the NGD central frequency and values shift between the simulated and measurement results is due to the tolerance of the considered dielectric substrate effective permittivity. Moreover, similar to all microwave circuits, the 0IO NGD specifications must include the insertion and reflection losses. As seen in Fig. 11(b), the NGD 0IO prototype presents maximal attenuation of only 2.5 dB around the NGD center frequency. The S-parameters plotted in Fig. 11(c) confirm that the reflection losses are better than 12 dB in the NGD BWs.

2) DISCUSSION ON NGD TEST RESULTS OF ROGERS-BASED 0IO PROTOTYPE
A good correlation between simulated and measured results is realized with Rogers substrate based ''0IO'' prototype introduced in Fig. 3(b) and Fig. 3(d). Once again, it can be seen that the circuit generates the dual-band NGD effect with center frequencies of about f 1 = 2.71 GHz and f 2 = 3.58 GHz as shown in Figs. 13(a).
Despite the slight differences indicated in Table 4, the transmission and reflection coefficients as reported in Fig. 13(b) and in Fig. 13(c), respectively are also in very good agreement notably in the NGD bandwidth. The same as in the results of previous case based on Figs. 11, the slight differences between simulated and measured results plotted in Figs. 13(c) are mainly caused by the imperfections of simulated and NGD circuit prototype fabrications.
To point out the innovative advantages of the 0lO structures, comparative study on NGD performance is addressed in the next subsection.

D. DISCUSSION ON NGD PERFORMANCES
The NGD passive performances of tested 0IO structures are, particularly interesting, compared to the existing dualband NGD circuits proposed in [24]- [27]. Table 5 summarizes the comparison between the NGD, insertion and reflection loss performances including the dual-band NGD bandwidth. It can be pointed out that the 0IO-shape circuit allows to achieve a very good attenuation showing lowloss aspect under very good reflection coefficient. Moreover, compared to the NGD circuits designed in [24]- [27], the 0IO-shape circuit is merely built with a fully distributed circuit without using any lossy lumped element as resistor.

IV. CONCLUSION
An innovative NGD analysis of particular 0IO-shape PCB traces is investigated. The proposed topology consists originally of 3-CL with lateral side interconnect in feedback with lossy and delayed different TLs. The S-matrix model of this particular NGD topology is established. Then, the NGD analysis is presented by exploiting the GD through the transmission coefficient expression. The NGD theoretical approach was validated by designing fully distributed microstrip structures without lumped elements even a resistance. Therefore, an NGD bandpass behavior is realized. A very good agreement between the simulations and measured S-parameters and GDs are realized.
The investigated ''0IO'' topology is promising for future RF and microwave system applications with the possibility to operate with low attenuation and reflection losses. This solution is utmost importance for the electronic systems designers, ensuring the system functioning jointly with PCB layout guidelines. BLAISE RAVELO (Member, IEEE) is currently a University Full Professor with NUIST, Nanjing, China. His research interest is on multiphysics and electronics engineering. He is also a Pioneer of the Negative Group Delay (NGD) concept about t < 0 signal travelling physical space. This extraordinary concept is potentially useful for anticipating and prediction all kind of information. He was a Research Director of nine Ph.D. students (seven defended), Postdoctoral Researchers, research engineers, and Master internships. With USA, Chinese, Indian, European, and African partners, he is also actively involved and contributes on several international research projects (ANR, FUI, FP7, INTERREG, H2020, Euripides 2 , Eurostars. . . ). He is a member of IET Electronics Letters Editorial Board as circuit and system subject editor. He has been a member of scientific technical committee of Advanced Electromagnetic Symposium (AES), since 2013. His Google scholar H-index in 2020 is 20. He is a member of research groups: the IEEE, URSI, GDR Ondes, Radio Society and coauthors of more than 250 scientific research articles in new technologies published in international conferences and journals. He is a Lecturer on circuit and system theory, science, technology, engineering, and maths (STEM) and applied physics. He is regularly invited to review articles submitted for publication to international journals the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON CIRCUITS  AND SYSTEMS, the IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY,  the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, IEEE ACCESS, IET CDS, and IET MAP, and books (Wiley and Intech Science). VOLUME 8, 2020