Member, IEEE Blaise Ravelo
email: blaise.ravelo@nuist.edu.cn

Member, IEEE Sébastien Lalléchère
email: sebastien.lallechere@uca.fr

Member, IEEE Wenceslas Rahajandraibe
email: wenceslas.rahajandraibe@im2np.fr

Member, IEEE T Fayu Wan
email: fayu.wan@nuist.edu.cn

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Keywords: Bandpass negative group delay (NGD), 3-D Electromagnetic cavity, EMI, Equalization method, Equivalent circuit, Synthesis equation

This paper deals with an original equalization method of 3-D electromagnetic (EM) cavity resonance by using bandpass negative group delay (NGD) function. Based on the equivalent circuit, the S-matrix model of the equalized EM cavity is introduced. The design equations of the NGD function are established in function of the cavity physical parameters. The feasibility of the equalization method is validated with a 42×28×3.8 cm rectangular cavity containing two communicating monopole antennas. By considering the TE110-mode, a significant equalization is realized with improvement of transmission coefficient flatness from 4-dB to less than 0.3-dB. In addition to the magnitude, the NGD equalization method enables to outstandingly reduce spikes of 3-D EM cavity group delay. With the proof of concept, the GD is reduced from more than 4-ns into 1-ns.

I. INTRODUCTION

HE performances of modern electronic and communication devices suffer from unintentional radiated electromagnetic interreferences (EMIs) [START_REF] Zhang | Short-Delay Multipath Mitigation Technique Based on Virtual Multipath[END_REF][START_REF] Kim | Performance Analysis of Distributed Single Carrier Systems With Distributed Cyclic Delay Diversity[END_REF][START_REF] Gao | Effect of Group Delay on Channel Estimation Performance in OFDM System[END_REF][START_REF] Mariappan | Effects of electromagnetic interference on the functional usage of medical equipment by 2G/3G/4G cellular phones: A review[END_REF][START_REF] Ishida | Evaluation of Electromagnetic Fields in a Hospital for Safe Use of Electronic Medical Equipment[END_REF][START_REF] Hanada | Possible electromagnetic interference with electronic medical equipment by radio waves coming from outside the hospital[END_REF][START_REF] Azaro | A semi-analytical approach for the evaluation of radiated immunity on a printed-circuit board in metallic enclosures[END_REF]. The interference issues were found at different scale of electronic systems. For example, the wireless system quality of services is frequently victim of propagation channel multipaths [START_REF] Zhang | Short-Delay Multipath Mitigation Technique Based on Virtual Multipath[END_REF], cyclic delay [START_REF] Kim | Performance Analysis of Distributed Single Carrier Systems With Distributed Cyclic Delay Diversity[END_REF] and group delay (GD) mitigation [START_REF] Gao | Effect of Group Delay on Channel Estimation Performance in OFDM System[END_REF]. The EMI effects were emphasized in different systems as medical equipment's [START_REF] Mariappan | Effects of electromagnetic interference on the functional usage of medical equipment by 2G/3G/4G cellular phones: A review[END_REF][START_REF] Ishida | Evaluation of Electromagnetic Fields in a Hospital for Safe Use of Electronic Medical Equipment[END_REF][START_REF] Hanada | Possible electromagnetic interference with electronic medical equipment by radio waves coming from outside the hospital[END_REF].

More frequently, EMI effects decrease printed circuit board (PCB) performances due to the enclosure cavity resonances [START_REF] Azaro | A semi-analytical approach for the evaluation of radiated immunity on a printed-circuit board in metallic enclosures[END_REF][START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF]. The characterization of metallic cavity resonances [START_REF] Ngu | A Complete Model for Simulating Magnitude and Phase of Emissions from a DUT Placed Inside a GTEM Cell[END_REF][START_REF] Hansen | Sources of problems in the GTEM structure and recommended solutions[END_REF] by using TEM cells is one of the most attractive activities of electromagnetic compatibility (EMC) research engineers. But because of the EMC characterization test costs, during the two last decades more and more modelling and simulation approaches of metallic enclosure EM resonances were developed [START_REF] Wang | Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes[END_REF][START_REF] Luo | Prediction of the Electromagnetic Field in Metallic Enclosures Using Artificial Neural Networks[END_REF][START_REF] Rajamani | Validation of Modal / MoM in Shielding Effectiveness Studies of Rectangular enclosures with Apertures[END_REF][START_REF] Lu | An efficient SPICE-compatible cavity resonant model for microstrip lines[END_REF][15][16][17][18][START_REF]EM/EMC Simulation Software[END_REF][START_REF] Robinson | Analytical formulation of the shielding effectiveness of enclosures with apertures[END_REF][START_REF] Yang | Coupling onto Wires Enclosed in Cavities with Apertures[END_REF][START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF][START_REF] Backstrom | Analytical model for boundary estimates of shielding effectiveness of complex resonant cavities[END_REF][START_REF] Gronwald | Method of moment analysis of a dipole antenna within a rectangular cavity[END_REF][START_REF] Gronwald | Calculation of mutual antenna coupling within rectangular enclosures[END_REF][START_REF] Lange | Broadband circuit model for electromagneticinterference analysis in metallic enclosures[END_REF][START_REF] Lange | Experimental validation of a broadband circuit model for electromagnetic-interference analysis in metallic enclosures[END_REF]. The successful popularity of computeraided design tools favors the development of algorithmic and iterative solvers for the modelling of EM cavities [START_REF] Wang | Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes[END_REF][START_REF] Luo | Prediction of the Electromagnetic Field in Metallic Enclosures Using Artificial Neural Networks[END_REF][START_REF] Rajamani | Validation of Modal / MoM in Shielding Effectiveness Studies of Rectangular enclosures with Apertures[END_REF]. The emergence of circuit and 3-D EM full wave commercial tools [START_REF] Lu | An efficient SPICE-compatible cavity resonant model for microstrip lines[END_REF][15][16][17][18][START_REF]EM/EMC Simulation Software[END_REF] was helpful for the EMC engineers to simulate certain EMC effects as the cavity-based structures. However, the simulation results of commercial tools are not always easy to interpret for many engineer users. Therefore, EMC cavity effect analytical formulations [START_REF] Robinson | Analytical formulation of the shielding effectiveness of enclosures with apertures[END_REF][START_REF] Yang | Coupling onto Wires Enclosed in Cavities with Apertures[END_REF][START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF][START_REF] Backstrom | Analytical model for boundary estimates of shielding effectiveness of complex resonant cavities[END_REF][START_REF] Gronwald | Method of moment analysis of a dipole antenna within a rectangular cavity[END_REF][START_REF] Gronwald | Calculation of mutual antenna coupling within rectangular enclosures[END_REF][START_REF] Lange | Broadband circuit model for electromagneticinterference analysis in metallic enclosures[END_REF][START_REF] Lange | Experimental validation of a broadband circuit model for electromagnetic-interference analysis in metallic enclosures[END_REF][START_REF] Nie | Accurate Modeling of Monopole Antennas in Shielded Enclosures with Apertures[END_REF][START_REF] Nešić | Shielding Effectiveness Estimation by using Monopole-receiving Antenna and Comparison with Dipole Antenna[END_REF] were developed in parallel for the mathematical understanding of EMC performances. Different approaches were suggested for the modelling of EM shielding for example by using coupling with metallic wires [START_REF] Yang | Coupling onto Wires Enclosed in Cavities with Apertures[END_REF] or by placing a receiver antenna [START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF][START_REF] Nešić | Shielding Effectiveness Estimation by using Monopole-receiving Antenna and Comparison with Dipole Antenna[END_REF] enclosed in the cavity. So far, these modelling methods enable to predict with significant accuracy the EM field inside the shielding enclosure under computation speed generally faster than the commercial tools and/or measurements.

Nevertheless, most of enclosure EMC problems cannot be completely solved with the simulators despite the success of popular full wave commercial tools [15][16][17][18][START_REF]EM/EMC Simulation Software[END_REF]. To face up the issues related to the EMC resonances emanated by cavity enclosures, few tentative techniques of resonance suppression were initiated in [START_REF] Jiao | Resonance suppression and electromagnetic shielding effectiveness improvement of an apertured rectangular cavity by using wall losses[END_REF][START_REF] Mahmood | Lossy resonators for suppressing power-bus resonance of printed circuit boards[END_REF][START_REF] Shahparnia | Electromagnetic interference (EMI) reduction from printed circuit boards (PCB) using electromagnetic bandgap structures[END_REF]. An improvement of shielding effectiveness was presented in [START_REF] Jiao | Resonance suppression and electromagnetic shielding effectiveness improvement of an apertured rectangular cavity by using wall losses[END_REF] by using a coating well structure. An introduction technique of lossy resonators is proposed in [START_REF] Mahmood | Lossy resonators for suppressing power-bus resonance of printed circuit boards[END_REF] to suppress the PCB power bus resonance. Then, a PCB EMI reduction technique was developed [START_REF] Shahparnia | Electromagnetic interference (EMI) reduction from printed circuit boards (PCB) using electromagnetic bandgap structures[END_REF] by using EM bandgap structures. However, those resonance suppression techniques are limited to very narrow bandwidth and requires quite expensive design and fabrication of shielding structures. Furthermore, the mechanism of these techniques is particularly hard to translate analytically.

To overcome these limitation challenges, a completely original technique of 3-D EM cavity resonance effect by using the unfamiliar bandpass negative group delay (NGD) function is initiated in the present paper. The basic principle of NGD equalization based on the cascading with positive group delay (PGD) circuit as a transmission line (TL) was introduced briefly in [START_REF] Ahn | Group Delay Equalized UWB InGaP/GaAs HBT MMIC Amplifier using Negative Group Delay Circuits[END_REF][START_REF] Ravelo | Synthesis of frequency-independent phase shifters using negative group delay active circuit[END_REF][START_REF] Broomfield | Broadband negative group delay networks for compensation of microwave oscillators and filters[END_REF]. The algebraic principle is ideally the addition of PGD and NGD that should generate GD equal to zero. The NGD equalization method was exploited to design a broadband phase shifter with constant phase independently to the frequency [START_REF] Ravelo | Synthesis of frequency-independent phase shifters using negative group delay active circuit[END_REF], for compensation of microwave oscillators and filters [START_REF] Broomfield | Broadband negative group delay networks for compensation of microwave oscillators and filters[END_REF], for expansion of bandwidth of tunable dispersion compensator based on ring resonators [START_REF] Takahashi | Expansion of bandwidth of tunable dispersion compensator based on ring resonators utilizing negative group delay[END_REF], and to design equalripple filter [START_REF] Qiu | A filter with equal-ripple negative group delay[END_REF]. A typical signal integrity exploitation of the NGD equalization technique was proposed in [START_REF] Ravelo | Neutralization of LC-and RC-Effects with Left-Handed and NGD Circuits[END_REF] to neutralize the RC-and LC-interconnect effects. But, so far, such a type of NGD application is strictly limited to planar RF and microwave circuits. In other word, in the best of the authors knowledge, because of its counterintuitive interpretation, the NGD function remains an unfamiliar concept for EMC engineers and no application is invoked. To break this roadblock, we expect that the unfamiliar NGD function should be a good candidate against EMC problems as the cavity resonance effect. Before the elaboration of this original idea, it is worth to describe quickly a recent state of the art on the bandpass NGD function.

Recent studies confirm the feasibility of the NGD microstrip and lumped circuit designs [START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF][START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF][START_REF] Ravelo | Active Microwave Circuit with Negative Group Delay[END_REF][START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF]. With passive circuit, the apparition of NGD effect is naturally accompanied by absorption [START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF] and/or reflection. To face up this constraint, challenging design of low-attenuation NGD circuits [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF] was achieved. In addition to loss reduction, compact NGD circuits were also designed [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF]. In contrary to the recursive questions about the existence of NGD topologies, the NGD function can be generated with any diverse electronic circuits, as presented in [START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF], by means of interference technique, an NGD circuit was proposed. To compensate the losses, active NGD circuits, for example, based on field effect transistor (FET) is proposed in [START_REF] Ravelo | Active Microwave Circuit with Negative Group Delay[END_REF]. Despite this progressive design, the NGD function remains an unfamiliar aspect for most of EMC engineers and requires further illustration for the basic understanding. In this way, an alternative approach illustrating the analogy between the NGD and filter functions was proposed in [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF]. Depending on the NGD frequency band positioning, the concept of low-pass and bandpass NGD functions was introduced [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF]. In the present paper, we are exploiting the bandpass NGD function to deal with the cavity EM resonance.

The paper is organized in four main sections. Section II introduces the basic principle of the resonance effect equalization by means of S-matrix modelling of linear timeinvariant (LTI) system. Section III develops the analytical equations of the cavity resonance suppression with bandpass NGD function. The design and synthesis equations of the NGD equalizer are established with respect to the resonator circuit parameters. Section IV is focused on the validation results. Then, Section V is the final conclusion.

II. ANALYTICAL THEORY OF 3-D EM CAVITY RESONANCE SUPPRESSION

This section describes the analytical mechanism of 3-D EM resonance suppression. After the specification of resonant system, the equations for calculating the bandpass NGD circuit parameters will be established.

A. Formulation of Cavity Resonance Problem

The cavity EM resonance effects was investigated regularly by electronic engineers with metallic cavity enclosures [START_REF] Azaro | A semi-analytical approach for the evaluation of radiated immunity on a printed-circuit board in metallic enclosures[END_REF][START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF][START_REF] Ngu | A Complete Model for Simulating Magnitude and Phase of Emissions from a DUT Placed Inside a GTEM Cell[END_REF][START_REF] Hansen | Sources of problems in the GTEM structure and recommended solutions[END_REF][START_REF] Wang | Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes[END_REF][START_REF] Luo | Prediction of the Electromagnetic Field in Metallic Enclosures Using Artificial Neural Networks[END_REF][START_REF] Rajamani | Validation of Modal / MoM in Shielding Effectiveness Studies of Rectangular enclosures with Apertures[END_REF][START_REF] Lu | An efficient SPICE-compatible cavity resonant model for microstrip lines[END_REF][START_REF] Robinson | Analytical formulation of the shielding effectiveness of enclosures with apertures[END_REF][START_REF] Yang | Coupling onto Wires Enclosed in Cavities with Apertures[END_REF][START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF][START_REF] Backstrom | Analytical model for boundary estimates of shielding effectiveness of complex resonant cavities[END_REF][START_REF] Gronwald | Method of moment analysis of a dipole antenna within a rectangular cavity[END_REF][START_REF] Gronwald | Calculation of mutual antenna coupling within rectangular enclosures[END_REF][START_REF] Lange | Broadband circuit model for electromagneticinterference analysis in metallic enclosures[END_REF][START_REF] Lange | Experimental validation of a broadband circuit model for electromagnetic-interference analysis in metallic enclosures[END_REF][START_REF] Nie | Accurate Modeling of Monopole Antennas in Shielded Enclosures with Apertures[END_REF][START_REF] Nešić | Shielding Effectiveness Estimation by using Monopole-receiving Antenna and Comparison with Dipole Antenna[END_REF][START_REF] Jiao | Resonance suppression and electromagnetic shielding effectiveness improvement of an apertured rectangular cavity by using wall losses[END_REF][START_REF] Mahmood | Lossy resonators for suppressing power-bus resonance of printed circuit boards[END_REF][START_REF] Shahparnia | Electromagnetic interference (EMI) reduction from printed circuit boards (PCB) using electromagnetic bandgap structures[END_REF]. To represent the resonance problem, we consider the rectangular metallic enclosure inspired from the demonstrator introduced in [START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF]. This enclosure is assumed defined by dimensions (Lx,Ly,Lz) containing two-monopole antennas with physical lengths d1 which are placed at M1(x1,y1=Ly/2,0) and M2(d+x1,y1,0). These two antenna terminals are supposed connected to two-port vector network analyzer (VNA) for Sparameter extraction:

( ) = ( ) ( ) ( ) ( ) (1)
with jω the complex angular frequency variable and the angular frequency variable 2 f = ω π . This metallic cavity is connected to the reference ground of the VNA. We assume that the monopole antennas are not connected to the cavity, they are linked to ports and . V jω and 2 ( ) V jω the spectra of signals at points M1 and M2, referred as ports and , respectively, the resonant cavity transfer function (TF) can be defined by:

2 1 ( ) ( ) / ( ) EMI T j V j V j = ω ω ω
(2) with the magnitude and phase written as:

( ) ( ) ( ) arg ( ) EMI EMI EMI EMI T T j T j  =     =     ω ω ϕ ω ω . ( 3
)
The associated GD is analytically defined by:

( ) ( ) / EMI EMI GD = -∂ ∂ ω ϕ ω ω .
(4) To analyze the cavity resonance response, the TF system is assumed to operate in the frequency band with center 0 0 / (2 )

f = ω π over bandwidth / (2 ) f ∆ = ∆ω π defined by: 2 1 ∆ = - ω ω ω (5) with: min 1 2 ( ) ( ) EMI EMI T T j T j = = ω ω . (6)
For example, at ω0, the dispersion system TF present the magnitude max 0 0 ( ) ( )

EMI EMI T T T j = = ω ω and GD max 0 ( ) EMI GD GD =
ω . The resonance problem can be graphically illustrated by typical "Λ" shape curve of magnitude and GD behavior shown in Fig. 2(c) and Fig. 2(d), respectively.

B. Topological Solution of the Cavity Resonance Problem

The principle of the BP NGD equalization of the previously introduced 3-D EM cavity will be described in the following paragraph.

1) S-Matrix Description of the Cavity Effect Equalization

Fig. 3 gives an overview of the problem statement, considering free-space antennas communication (Fig. 3(a)), presence of a cavity (Fig. 3(b)), and use of BP NGD system (Fig. 3(c)). Thus, Fig. 3(a) depicts the block diagram of the dipole antennas communicating in free-air. The dipoles can be assumed as a two-port system connected to ports and . , ω ω defined in Figs. 2(c) and 2(d),

this reference system can be modelled with S-matrix: 0 ( )

0 air S j     ≈       ξ ω ξ (7)
with ξ is the transmission loss assumed to be constant and frequency independent. With well-matched antennas placed inside the cavity as shown in Fig. 3(b), we can consider that the resonant system is equivalent to the S-matrix:

11 21 21 11 ( ) ( ) ( ) ( ) ( ) EMI EMI EMI EMI EMI S j S j S j S j S j     =       ω ω ω ω ω . ( 8
)
The main problem of the present study is to find a topological solution allowing to generate S-matrix, in the frequency band

[ ] 1 2
, ω ω :

11 12 21 22 ( ) ( ) ( ) ( ) ( )
eq eq eq eq eq S j S j S j S j S j

    =       ω ω ω ω ω (9)
with reflection and isolation coefficients:

11,12,22 ( ) 0 eq S j ≈ ω [START_REF] Hansen | Sources of problems in the GTEM structure and recommended solutions[END_REF] and transmission coefficient independent to frequency:

21 ( ) 1 eq S j ≈ ω . ( 11
)
We emphasize that:

• The term isolation coefficient here, is related to the use of unilateral active circuit, for example, using FET. In the ideal case of such circuit, S12 is close to zero. • The justification of relation ( 11) is due to the fact that we want to restitute the distorted signal in order to generate an output as similar as possible to the corresponding input.

2) Justification of Bandpass NGD Function Equalizer Against the EMI Effect

The proposed analytical solution consists analytically in identifying an equalizer transmission coefficient (13) This unknown S-matrix can be identified by considering the resonant characteristic of the cavity responses introduced in Figs. 2(c) and 2(d). Therefore, the corresponding magnitude and GD must verify, respectively:

? 21 21
( ) 1 / ( ) 1 EMI S j S j ≈ > ω ω ( 14
) ? 21 ? arg ( ) ( ) ( ) 0 EMI S j GD GD   -∂   = ≈ - < ∂ ω ω ω ω . ( 15
)
This last equation explains that the unknown system solution operates with NGD function. For this reason, the topological solution of block diagram illustrated in Fig. 3(c) is developed in the present paper. However, it is noteworthy that by adopting solution implemented with equation (15), the phase, 21 21 ( ) arg ( )

eq eq S j   =  
ϕ ω ω , is not equal to zero. However, in this case, we still can realize the equalization when the magnitude and GD of the EMI effect are reduced to be closed to zero. In general, the NGD S-matrix can be written as:

? 11 21 21 11 ( ) ( ) ( ) ( ) ( ) ( ) N N N N N S j S j S j S j S j S j       = =         ω ω ω ω ω ω . ( 16
)
This topological solution expects to operate with best results if ports and are well-matched:

11 22 11 22 ( ) ( ) 1 ( ) ( ) 1 EMI EMI N N S j S j S j S j  = <<   = <<   ω ω ω ω . ( 17
)
3) Specifications of Bandpass NGD Function As initiated in [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF], a bandpass NGD function is specified by:

• The NGD center frequency, /

n n f = ω π ,

• The NGD value, ( ) min ( ) 0

N N N n n GD GD GD   = = <   ω ω ,
• And the NGD bandwidth ( ) 0

N N GD = ω . ( 18
)
As reported in [START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF], the bandpass NGD function behaves similar to the bandpass filter but the analysis is not relying on the gain. We must focus on the behavior of:

21 arg ( ) ( ) N N S j GD   -∂   = ∂ ω ω ω . ( 19
)
In graphical representation, the BP NGD responses related to the magnitude and GD behave generally as ideal curves as sketched in Fig. 4(a) and Fig.

III. NGD DESIGN EQUATIONS FROM EQUIVALENT CIRCUIT ANALYSIS

Based on the general solution evoked in the previous section, design equations of the NGD circuit are established in function of the cavity structure parameters.

A. Equivalent Circuit Description

For the sake of analytical simplicity and better understanding, we adopt the following hypotheses:

• The internal impedances of dipole1 and dipole2 are neglected, • Only one of TEmn or TMmn propagative modes is considered,

• And the FET equivalent circuit is constituted by its transconductance gm and drain-source resistance Rds. The proposed assumptions are relevant with the current work. First, the antennas internal impedances are weak considering Hertzian dipole, compliant with the equivalent circuit model [START_REF] Ragan | Microwave transmission circuits[END_REF] used in this work. Then, for the sake of simplicity, the methodology is presented with a unique propagative mode. These results might be extended to multimodal propagation. Finally, the proposed FET equivalent structure (i.e. with transconductance and drain-source resistance) is classically met, e.g. for NGD applications [START_REF] Ravelo | Active Microwave Circuit with Negative Group Delay[END_REF].

Based on the EM cavity theory, the 3-D structure introduced in Fig. 3(b) is equivalent to the circuit shown in Fig. 5(a). The model is relying on the developments proposed in [START_REF] Ragan | Microwave transmission circuits[END_REF] for the modeling of EM cavities. The principles were applied in [START_REF] Moussa | Simulation of a 2D cavity under Qucs[END_REF] with electrical simulator tools to the simulation of multidimensional mode stirred reverberation chambers. The access ports with subscripts k=1,2 are connected to transformers Tk(nk:1) which are specified by transform ratio, nk. The considered propagative mode is equivalent to the RdLdCdparallel network mounted in parallel. The resonance frequency is given by:

2 2 2 0 y x z x y z m m m c L L L       ≈ + +               ω π . ( 20
)
with mx,y,z are integers corresponding to the propagation modes, TE/TM(mx,my,mz). The equivalent circuit of equalized cavity proposed in Fig. 3(c) is depicted in Fig. 5(b). It is composed of the cavity equivalent circuit (Fig. 5(a)) in cascade with the NGD active cell. This later one consists of an RLC-series network mounted in parallel connected between the gate and source of an FET with drain and source connected by a shunt resistance Rm.

Based on the principle of the BP NGD equalization explained in the previous section, the design method of NGD solution in function of the 3-D EM cavity will be described in the following subsection via S-matrix modelling of equalized circuit presented in Fig. 5(b).

B. S-Matrix Model of the NGD Equalized Circuit

By denoting the Laplace variable, s j = ω , the two-port S- matrix model, ( ) eq S s     , of the NGD equalized circuit presented in Fig. 5(b) can be determined from the ABCD matrix, ( )

eq ABCD s     .
Because of the cascade configuration, we have:

( ) ( ) ( ) eq EMI N ABCD s ABCD s ABCD s       = ×       (21)
where:

• The cavity equivalent circuit ABCD matrix is written as:

1 2 1 1 2 2 1 0 ( ) ( ) EMI d n n ABCD s n Y s n n n         =         (22)
• The NGD equivalent circuit ABCD matrix is given by:

1 1 ( ) 1 ( ) ( ) m ds m ds N m ds m m ds R R R R ABCD s R R g R R Z s Z s +     -    =   +       (23)
where:

1 1 ( ) ( ) 1/ ( ) d d d d Y s C s R L s Z s R Ls Cs  = + +    = + +  . ( 24
)
By means of ABCD-to-S matrix transform, with the reference impedance, R0=50 Ω, we have equalized cavity S-matrix elements:

2 2 1 0 2 11 2 2 1 0 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) d eq d Z s n R n Y s Z s S s Z s n R n Y s Z s   - +   =   + +   (25) [ ] { } 1 2 0 21 0 2 2 1 0 2 2 ( ) ( ) ( ) ( ) ( ) ( ) eq m ds m ds m ds d n n R g R Z s S s R R R R R Z s n R n Y s Z s - = + +   + +   (26) 0 22 0 ( ) ( ) ( ) eq m ds m ds m ds m ds R R R R R S s R R R R R - + = + + . ( 27
)

C. Design Equation of the NGD Circuit Equalizer

As introduced in Section II, the initial data of the problem corresponds to the 3-D EM cavity parameters. The solution of the problem is the NGD circuit design. The next paragraphs develop how the NGD design equations are established.

1) Equalization Hypothetical Equation

The design equation of the NGD circuit equalizer are the expression of the parameters Rm, R, L and C in function of the EM cavity ones Rd, Ld and Cd.

d d C L = ω . ( 28
)
The elaboration of the equations is based on the hypotheses:

• the NGD center frequency should coincide with the cavity resonance frequency:

0 1 / N n LC = = ω ω . ( 29
) by inverting this equation, we can write the capacitor value:

2 0 1/ ( ) C L = ω . (30)
• The equalized cavity input/output reflection and transmission coefficients obey equations ( 10) and [START_REF] Wang | Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes[END_REF].

2) Synthesis Equations of R and L

The magnitude of the transmission coefficient given in equation ( 26) is equal to:

[ ] 2 2 2 2 2 2 1 2 0 0 21 2 2 0 2 ( ) ( ) ( ) ( ) ( ) m ds eq m ds m ds n n R g R R L S R R R R R X Y + - = + + + ω ω ω ω ω ω (31)
with:

2 2 2 2 0 1 0 0 0 ( ) ( ) ( ) d d d X n R RR L L R R   = - + +   ω ω ω ω ω (32) 2 2 2 1 0 2 2 4 4 0 1 0 (2 ) ( ) ( ) d d d d d d L R R n R L R R L n L Y R R Ln       + + +       =   - +     ω ω ω ω . ( 33
)
The Taylor series limited expansion of this transmission coefficient

eq eq S S R L O     ≈ + - +     ɶ ω ω ψ ω ω ω ω (34)
where:

[ ]

1 2 0 21 0 2 2 0 1 0 2 2 ( ) ( ) ( ) eq m d m ds m ds m ds d d n n R R R g R S R R R R R R R n R R n R =   + + + +   ω (35) [ ] 4 2 3 2 3 2 2 4 2 2 4 1 2 0 4 0 2 0 1 0 0 2 2 4 0 0 4 2 2 1 0 2 16 ( ) ( ) ( ) ( ) m d m ds d d m ds m ds d d n n R R R g R n R R R L R R R R R R R n R R n R + - = + +   + +   ψ ω ψ ω ψ ω ω (36)
with:

2 2 2 2 1 2 0 2 2 2 2 2 4 2 1 0 2 2 (2 ) 2 d d d d d n n R R R L L n L L n R n R R R R  =     = + +     ψ ψ . ( 37
)
Consequently, equation ( 11) can be rewritten as:

S R L  ≈   ≈   ω ψ ω . ( 21 0 0 ( ) 1 ( , , ) 0 eq
) 38
The solutions of this equation system are the NGD equalizer parameters expressed as:

2 2 1 2 0 2 0 1 1 0 2 ( ) d d m ds d m ds m ds n n R R R n R R g R n n R R R R R R =   - -   + +   (39) 2 2 1 0 2 2 1 0 1 2 0 2 2 2 2 0 1 0 2 2 ( ) 2 ( 2 )
d d d d d d d R R n R n R R n R R R n n R R L n L R R n R n R R     + +         -   =   + +   ω . ( 40
)
The existence condition of the resistance related to the FET parameters is defined by:

[ ] min 2 0 1 0 1 2 0 ( ) ( ) 2 d m ds m ds m m m d ds R n R R R R R R g g n n R R R R + + + ≥ = . ( 41
)

3) Synthesis Equations of Rm

The resistance Rm serve to the NGD circuit output matching. In function of the matching constraint r (for example better than -10 dB), it can be determined from equation:

)

ds ds ds m ds ds ds r R R r R R R R R r R R r R R R R -   + + -  =  +   + + -  . ( 43
)
To verify the feasibility of the developed equalization method, a proof-of-concept will be examined in the next section.

IV. PROOF-OF-CONCEPT OF CAVITY RESONANCE EQUALIZATION

The present section deals with the validations of the developed NGD equalization method for minimizing the effect of 3-D EM cavity resonance. The numerical applications in both frequency and time domains were performed in Matlab environment.

A. Frequency Domain Validation of the EM Cavity NGD Equalization Method

After the proof-of-concept, description, validation results based on transmission coefficient and GD are presented in this subsection.

1) Proof-of-Concept Description

The proof-of-concept circuits of the 2-D lower resonance frequency of the rectangular cavity designed in the ADS® schematic environment are presented in Fig. 6. Following works in [START_REF] Ragan | Microwave transmission circuits[END_REF][START_REF] Moussa | Simulation of a 2D cavity under Qucs[END_REF], the proof-of-concept physical and electrical parameters are addressed in Table I. The resonance behavior of this structure was reported in [START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF]. The interested reader may find the equivalence between cavity parameters definition and equivalent circuit design, developed through Kron's methodology, in [START_REF] Leman | New approaches in Electromagnetic Compatibility: Use of the circuit approach to solve large EMC problems[END_REF]. The cavity is assumed to be constituted by Copper enclosure and wire antennas. The NGD equalizer parameters were calculated from equations ( 39), ( 40) and ( 43) by taking r=-10 dB. The minimal value of FET transconductance to realize the equalization is gm,min=22 mS.

2) Frequency Domain Demonstration of Equalized Cavity Resonance Responses

The present frequency domain analysis is based on the Sparameter simulated from 0.4 GHz to 1 GHz. The feasibility of the NGD equalization can be understood with resonant ("EMI" in red dashed line) and equalized ("Eq." in black solid line) of S21 and GD plotted in Figs. The EM cavity resonance effect manifests with concave response around f0 with peak S21(f0)=-0.72 dB and GD(f0)=4.47 ns. The dispersion due to the resonance over the bandwidth BW=0.1 GHz can be estimated with the ranges of attenuation ∆S21≈-6 dB and ∆GD≈4 ns. Thanks to the introduction of NGD circuit, a significant reduction of S21 and GD fastnesses into ∆S21NGD<-1 dB and ∆GDNGD<0.3 ns can be observed respectively with black curves of Fig. 7(a) and Fig. 7(b), respectively.

3) Equalized Results with Parametric Analysis with Respect to the Cavity Equivalent Circuit Parameters

In addition to the previous validation, for the better understanding on the NGD equalization method robustness, parametric analyses were performed with different values of Rd={0.1 to 100 kΩ}, Ld={0.9 to 1 nH} and Cd={50 to 90 pF}. Mapping S21 and GD results in the frequency band 0.5 to 0.8 GHz with respect to, Rd in logarithmic scale, is introduced in Figs. 8. It can be emphasized that the equalization effect is still confirmed by the improvement of flatness of (∆S21EMI, ∆GDEMI) compared to (∆S21eq., ∆GDeq.). This flatness reduction robustness is also verified with the other parametric analyses with respect to Ld and Cd as seen in Figs. 9 and in Figs. 10, respectively. However, in these two last cases of parametric analyses the cavity resonance frequencies are intentionally shifted while the NGD center frequency was kept fn=646 MHz. More rigorous illustration of the resonance cavity NGD equalization method is discussed in the following subsection by considering a Gaussian pulse modulated with sine wave signal carrier with f0.

B. Time Domain Results of Bandpass NGD Equalization

A single pulse signal presenting a Gaussian waveform with about 17 ns time width and 1 V amplitude was assumed as a test signal. The time domain analysis was performed in 40 ns time window with 0.25 ns time step. This input signal was injected to the cavity and equalized cavity circuits by taking into account the source and load impedances R0=50 Ω. Therefore, we obtain the transient input ("vin"), resonance circuit ("vEMI"), and equalized signal ("veq."), results respectively depicted in Figs. 11(a-c). The corresponding signal envelopes ("Env.") are shown in Figs. 12. Fig. 11 and Fig. 12 rigorously confirm the effectiveness of the bandpass NGD function equalization in the time-domain.

where the subscript, κ={in, EMI, eq.}. Accordingly, we have considered the correlation coefficient, xκ, between the initial and signal envelope subscripted, κ={EMI, eq.}:

{ } max max , , 1 2 , , 1 ( ) ( ) ( ) ( )
k in in ave ave k k in in ave ave k v k v v k v x v k v v k v = = - -         = - -         ∑ ∑ κ κ κ κ κ . ( 45
)
The calculated cross correlation between the input and EMI distorted signals plotted in Figs. 12 was equal to xEMI=66.2%.

After the NGD equalization, the cross correlation was widely improved to xeq.=99.5%.

V. CONCLUSION

An original equalization method of 3-D EM cavity EMI and resonance effect by using bandpass NGD function is initiated. The developed NGD method presents a considerable simplicity. The method is based on the S-matrix modelling of the cavity and NGD equivalent circuit. The equalization analytical mechanism as a function of the NGD key parameters is described. The NGD function synthesis approach is established as a function of the EMI effect.

The relevance of bandpass NGD equalization is illustrated with both frequency and time domain applications. A cavity proof-of-concept corresponding to a rectangular EM enclosure was performed. The frequency and time domain analyses were realized by considering equivalent circuits simulated in a commercial tool environment. The NGD equalization feasibility was quantified with improvement of transmission coefficient and GD flatness around the resonance frequency. Furthermore, time domain analyses rigorously confirm the feasibility of the EMI equalization effect. The distorted signal envelope was clearly improved with the correlation coefficient increasing from 66.2% to more than 99.5%.

As ongoing research, the proposed NGD EMI equalization method will be validated by measurement and also applied to multiple EMI source effect reduction. In the future, the developed NGD equalization method can be potentially exploited to correct complex EMIs as in the reverberation chamber (RC) [START_REF] Magdowski | Coupling of stochastic electromagnetic fields to a transmission line in a reverberation chamber[END_REF][START_REF] Tarhini | Susceptibility of printed circuit boards in complex electromagnetic environment[END_REF]. From both simulation and experimental point of views, the proposed methodology (noise mitigation through NGD equalization) might also be helpful when dealing with EMC issues involving printed circuit boards (PCBs), as for numerical [START_REF] Tarhini | Susceptibility of printed circuit boards in complex electromagnetic environment[END_REF][START_REF] Li | Combined FEM-reaction method for EMS analysis of a PCB in a low-Q enclosure[END_REF][START_REF] Leman | Prediction of Electronic Unit Radiated Emissions from Both PCB Near Field and Shielding Enclosure Characterization[END_REF] EMC susceptibility and/or emission [START_REF] Li | Combined FEM-reaction method for EMS analysis of a PCB in a low-Q enclosure[END_REF] problems.

Fig. 1 .

 1 Fig. 1. Scenario of metallic enclosure with two communicating dipole antennas.

Fig. 2 .

 2 Fig. 2. Resonant system: (a) transient and (b) envelope responses, and TF (c) magnitude and (d) GD.

Fig. 2 (

 2 Fig.2(a) illustrates a black box response of the resonant enclosure. As result of EM resonance, the output v2 undergoes spreading effect compared to corresponding input v1, as illustrated by the increase of envelope pulse width as depicted in Fig.2(b). By taking 1 ( )V jω and 2 ( ) V jω the spectra of signals at points M1 and M2, referred as ports and , respectively, the resonant cavity transfer function (TF) can be defined by:

Fig. 3 .

 3 Fig. 3. Block diagram description of two-dipole antenna in (a) free-air, (b) inside a 3-D rectangular cavity and (c) in the cavity and dipole2 connected to NGD circuit.

  S jω (the symbol "?" standing for the searched term) allowing to realize the operation: By supposing the equalized transmission coefficient of equation[START_REF] Wang | Convergence acceleration and accuracy improvement in power bus impedance calculation with a fast algorithm using cavity modes[END_REF], we can derive the solution of previous equation:

Fig. 4 .

 4 Fig. 4. NGD TF (a) magnitude and (b) GD responses.

Fig. 5 .

 5 Fig. 5. Equivalent circuits of (a) simple and (b) NGD equalized cavity.

Fig. 6 .

 6 Fig. 6. Simulated schematic of cavity and cavity+NGD equivalent circuits.

7 .

 7 The bandpass NGD function is shown by the dotted blue curves with NGD center frequency fn=f0 and NGD value about -0.85 ns.

Fig. 7 .

 7 Fig. 7. EMI (dashed red line), NGD (dotted blue line) and equalized (solid black line): (a) magnitude and (b) GD frequency responses.

Fig. 8 .

 8 Fig. 8. Mapping of (a) EMI and (b) equalized circuit transmission coefficients, and (c) EMI and (d) equalized circuit GD vs Rd.

Fig. 9 .

 9 Fig. 9. Mapping of (a) EMI and (b) equalized circuit transmission coefficients, and (c) EMI and (d) equalized circuit GD vs Ld.

Fig. 10 .

 10 Fig. 10. Mapping of (a) EMI and (b) equalized circuit transmission coefficients, and (c) EMI and (d) equalized circuit GD vs Cd.

Fig. 11 .

 11 Fig. 11. Transient plot results of (a) input, (b) EMI and (c) equalized signals.

Fig. 12 .

 12 Fig. 12. (a) Natural and (b) normalized envelope (env.) plot results of input, EMI and equalized signals.As expected in the previous frequency analysis, the EMI signal vEMI of Fig.11(b) illustrates the time spreading due to the resonance from the cavity. The coherence between the input and output signals can be assessed with correlation coefficient comparisons[START_REF] Mickelson | A comparison of two correlation schemes[END_REF]. This correlation can be estimated with discrete signals, for example, giving the number of sampling,

TABLE I CONSIDERED

 I PARAMETERS OF THE 3-D EM CAVITY

	Type	Description	Name	Value
			Lx	42 cm
		Length	d1	3 cm
	Geometrical		d2	2 cm
		Width	Ly	28 cm
		Height	Lz	3.8 cm
		Resonance frequency	f0=f1,1,0	646 MHz
	Cavity equivalent circuit	Resistance Inductance Capacitance Transformer coefficient	Rd Ld Cd n1	47 kΩ 0.93 nH 65.5 pF 1
		ratio	n2	0.7
		FET	gm Rds	70 mS 200 Ω
	NGD circuit	Resistance	Rm R	185 Ω 11 Ω
		Inductance	L	5.98 nF
		Capacitance	C	10 pF

Date of publication xxx xx, 2020. This research work was supported in part by NSFC under Grant 61601233 and Grant 61971230, in part by the Electrostatic Research Foundation of Liu Shanghe Academicians and Experts Workstation, Beijing Orient Institute of Measurement and Test under Grant BOIMTLSHJD20181003, in part by the Jiangsu Innovation and Enterprise Group Talents Plan 2015 under Grant SRCB201526, and in part by The Priority Academic Program Development of Jiangsu Higher Education Institutions.


Member, IEEE Blaise Ravelo
email: blaise.ravelo@nuist.edu.cn

Yajian Gan
email: yajian.gan@im2np.fr

Member, IEEE Fayu Wan
email: fayu.wan@nuist.edu.cn

Member, IEEE Wenceslas Rahajandraibe
email: wenceslas.rahajandraibe@im2np.fr

Member, IEEE Sébastien Lalléchère
email: sebastien.lallechere@uca.fr

Member, IEEE D Zhifei Xu

Resonance Effect Reduction with Bandpass Negative Group Delay Function

Keywords: Correction method, Bandpass negative group delay (NGD), Concave dispersion, Synthesis relation

come

I. INTRODUCTION

ESPITE the tremendous technological progress, the communication system performance suffers naturally of undesirable delay and resonance perturbations [START_REF] Hwang | Slope Interconnect Effort: Gate-Interconnect Interdependent Delay Modeling for Early CMOS Circuit Simulation[END_REF][START_REF] Groenewold | Noise and Group Delay in Active Filters[END_REF][START_REF] Wijenayake | RF Analog Beamforming Fan Filters Using CMOS All-Pass Time Delay Approximations[END_REF][START_REF] Kim | Performance Analysis of Distributed Single Carrier Systems With Distributed Cyclic Delay Diversity[END_REF][START_REF] Lei | Effect of Group Delay on Channel Estimation Performance in OFDM System[END_REF][START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF]. In fact, the degradations of transmission signal are intimately linked to the channel dispersion effect. Several constraints constitute physical factors limiting the electronic system performances. For example, the non-flat transmission gain and the propagation delay related to dispersive channels [START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF][START_REF] Zhang | Short-Delay Multipath Mitigation Technique Based on Virtual Multipath[END_REF] are major one challenging constraints of the communication system design. In addition, the time-varying propagation channel effects degrade considerably the performances of wireless network systems [START_REF] Li | Improved Switched System Approach to Networked Control Systems With Time-Varying Delays[END_REF]. Some of recent wireless systems were designed with multiagent concept and operating with varying time delays [START_REF] Xiong | Containment Control for Discrete-Time Multiagent Systems With Communication Delays and Switching Topologies[END_REF].

To face up to this issue of resonance and dispersion effects, an innovative method compared to the complex signal processing approaches is necessary. The present paper is focused on alternative correction method based on the bandpass negative group delay (NGD) function. The NGD equalization [START_REF] Ravelo | Neutralization of LC-and RC-Effects with Left-Handed and NGD Circuits[END_REF] was used and implemented with electronic circuits to neutralize group delay (GD) susceptible to be generated by electrical interconnection RC-and LC-circuit model.

In difference to most of the existing signal correction techniques against the channel dispersive degradation, the NGD method under investigation is implemented in both circuit and system approaches. The analytical implementation can be realized in both the frequency and time domains. Moreover, the NGD method presents an outstanding simplicity and flexibility to any dispersion frequency band. Before exploring the dispersion reduction method, it is worth slightly describing the unfamiliar NGD functions. It is noteworthy that the nonconventional NGD functions behave with mathematical analogy with the classical linear filter gain [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF]. Different categories of NGD functions as low-pass, high-pass and bandpass (BP) NGDs were classified. The corrector developed in this paper is originally based on the bandpass NGD function. This particular function attracted recently attention of radiofrequency (RF) and microwave engineers by designing different topologies of bandpass NGD circuits [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF][START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF][START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF]. Compact circuit and absorptive filter-based BP NGD circuits were designed [START_REF] Shao | A compact transmission line self-matched negative group delay microwave circuit[END_REF][START_REF] Qiu | Absorptive bandstop filter with prescribed negative group delay and bandwidth[END_REF]. The unfamiliar BP NGD function was also implemented with microwave signal interference effects [START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF][START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF]. Knowing the "V"-shape behavior [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF] regularly associated to the BP NGD magnitude, an original method enabling to reduce the resonance dispersion effect is developed in the paper.

This paper is organized in three main sections, complementary to this introductory part. Based on transfer function (TF) approach, Section II develops the analytical principle of the NGD method under study. The mechanism of the resonance dispersion reduction will be elaborated with mathematical equations including the NGD parameter formulation in function of the dispersion specifications. Section III is focused on the feasibility analysis. Proof-of-concept (POC) circuit will be analyzed with illustrative comparison of calculated and simulated magnitudes and GDs of dispersion and NGD TFs. Last, Section IV is the conclusion.

II. THEORY ON THE BANDPASS NGD METHOD OF RESONANCE DISPERSION REDUCTION

This section describes the circuit theory on the dispersion effect reduction. After the canonical definitions of the dispersion, the equivalent passive RLC circuit will be considered. Then, the design formulas of appropriated BP NGD circuit in function of the dispersion parameters will be originally established.

A. Specifications of Resonance Effect

The resonance effect under study is generated by the circuit constituted by Rd, Ld and Cd passive components depicted in Fig. 1(a). By taking V1(jω) and V2(jω) the spectrums of signals referred as ports  and , respectively, the circuit voltage TF can be expressed as: ( )

2 21 ( ) ( ) / ( ) 1 / ( 1) d d d d d T j V j V j L C s R C s = = + +    (
d d d d d d d d d d d d R C L C C R C L L C - + ==  - +        . ( (1 ) () 1 ( 2 )
) 2
The dispersion due to the resonance phenomenon can be represented by the TF magnitude and GD responses depicted in Figs. 1(b) and 1(c). The resonance is specified by the center frequency:

2 1/2 0 2 / ( 2 ) d d d d d L C L C =-  . ( 3
)
It can be found that the resonance frequency exists under the condition:

2 /2 d d d L R C  . ( 4
) At ω0, the TF presents the magnitude and GD, respectively: (8) The associated GD is defined, respectively, by: 00

2 2 2 0 0 ( ) 4 / (4 ) d d d d d d d T T L R C L R C = = -  . ( 5
) 00 ( ) 2 / d d d LR ==    . ( 6
( / 2) d - = -      . ( 9
)
The resonance dispersion TF can be rewritten in the canonical form:

1/ dd LC =  (11) / d d d R C L =  (12)
It can be originally demonstrated that knowing, the peak, T0, the resonance TF GD at ω0, is expressed as:

22 0 0 0 0 0 2 1( 1) / T T T = - + -  (13)
for any given resonance effect specified at frequency ω0, the Ld and Cd parameters can be synthesized by the equations:

22 0 0 0 0 1( 1) / ( 2 ) dd L R T T T = - + -  (14)
2 2 0 0 0 0 2 / 1( 1) d d C R T T T   = - + -      . ( 15
)

B. Identification of Appropriated Bandpass NGD Function

To reduce the resonance effect, we will employ the BP NGD function with properties and specifications introduced in this subsection. The bandpass NGD function [START_REF] Wan | O=O Shape Low-Loss Negative Group Delay Microstrip Circuit[END_REF] allows to operate in the signal frequency band where the GD presents a negative value. The NGD function is implemented with the passive cell composed of RLC-parallel network combined to resistance, R0 as shown in Fig. 2(a). The bandpass NGD function is specified, inversely to the resonance responses of Figs. 1(b) and 1(c), with magnitude and GD responses depicted in Figs. 2(b) and 2(c), respectively. The bandpass NGD function is defined by its NGD center frequency, ωn, and bandwidth, Δω. The BP NGD voltage TF canonical form can be written as:

0 2 2 2 2 ( ) ( ) / ( ) n n a n b n T s T s s s s = + + + +     (16)
where:

1/ n LC =  (17) 1 / ( ) a RC =  (18) 00 ( ) / ( ) b R R R RC =+ 
(19) At ω0, the associated magnitude and GD, which is defined by:  

( ) arg ( ) / n n Tj = -      (20)
are expressed as, respectively:

0 0 0 0 0 ( ) ( ) / / ( ) n n n a b T T j T R R R = = = +     (21) 22 0 0 0 ( ) 2( ) / ( ) 2 / ( ) n a b a b R L R R  = - = - +         . (22)
This last expression, ( ) 0 nn   , is always negative for any value of RLC circuit parameters. This remark confirms the passive cell introduced in Fig. 2(a) behaves as BP NGD function. This linear system can be originally characterized by NGD cut-off frequencies written as:

1/2 1 0 0 ( ) / ( ) nn x L R R x R R  = - +     (23) 1/2 2 0 0 ( ) / ( ) nn x L R R x R R  = + +     ( 24
)
with:

22 00 ( ) 2 n x L R R R R = + +  . ( 25
)

C. NGD Function Parameters Analytical Determination

To reduce the resonance effect illustrated in Figs. 1(b) and 1(c), the association of dispersion and BP NGD TFs seems to be an intuitive solution. The following paragraphs describe the analytical mechanism of the NGD function synthesis. The design feasibility of the canonical system introduced in ( 16) is confirmed by these NGD specifications.

1) Principle of Resonance Reduction

To realize the resonance effect reduction, the total TF magnitude and GD are expected to behave as presented in Fig. 3(a) and Fig. 3(b). It means that the associated TF must be close to unity T(s)≈1. Because of dispersion, the output of system introduced in Fig. 1(b-c) is distorted compared to input, V2≠V1.

To overcome this undesirable dispersion issue, the aim of correction consists in introducing a solution allowing to generate corrected output V 2 =T(V 1 )≈V 1 especially, in the frequency band [ω0,Δω]. The developed resonance reduction principle is aimed to introduce the BP NGD function. The correction technique under investigation consists originally in cascading the resonance and NGD circuits as illustrated in Fig. 3(c). Under the consideration of the inter-stage matching follower, the total circuit TF can be analytically modelled by:

T(s)=T d (s)×T n (s). (26)

2) NGD Parameters Synthesis Formulation in Function of Resonance Specifications

This paragraph describes the analytical mechanism of the dispersion correction NGD method. As target of the analysis, the NGD TF parameters must be determined in function of the resonance parameters introduced in Figs. 1. The resonance effect reduction means the annihilation of the total TF proposed in equation ( 26). In other words, by taking ωn=ω0, the corrected dispersion TF magnitude and GD must be close to:

0 0 0 ( ) ( ) ( ) 1 dn T T T =     ( 27
) 0 0 0 ( ) ( ) ( ) 0 dn = +        . ( 28
) Knowing the dispersion and NGD transmittances defined in equation [START_REF] Ravelo | Neutralization of LC-and RC-Effects with Left-Handed and NGD Circuits[END_REF] and equation ( 16), the dispersion effect reduction can be realized by synthesizing the NGD circuit parameters R, L and C knowing T0, ω0 and for the given resistance load, R0. It yields from equation ( 27), the inversion of equation Tn0=1/T0, and (21), the NGD parameter synthesis relation:

00 ( 1) R T R =- . ( 29
) It can be derived from equation (28) and the GD expressed in equation ( 22), the dispersion GD annihilation leads to the inductance synthesis equation:

  1/2 2 2 1/2 2 0 0 0 0 0 0 2 ( 1) / ( 1) 1 L R T T T T  = - - + -   . ( 30
)
By inversion of resonance frequency, ωn=ω0, of equation ( 17), we have the capacitor formula:

1/2 2 1/2 2 2 0 0 0 0 0 0 0 ( 1) 1 / 2 ( 1) C T T T T R T   = - + - -    . (31)

III. PROOF-OF-CONCEPT OF RESONANCE EFFECT REDUCTION

WITH BP NGD CIRCUIT The present section deals with the validation of the previous theoretical approach. A POC constituted by a lumped circuit generating the resonance effect reduction and corrected with a BP NGD function will be investigated based on the frequency domain or AC analyses. This validation study consists mainly in comparing the responses from the dispersion, NGD and both combined NGD-dispersions TFs expressed in (1), ( 16) and ( 27), respectively, calculated with Matlab and simulations.

A. Validation with POC Simulated Results

The dispersion parameters (T0, f0) were arbitrarily chosen to validate the NGD resonance effect reduction. The associated RLC resonant circuit inductance, Ld and capacitance, Cd, addressed in Table I, were calculated from equations ( 14) and ( 15) by fixing the value of resistor, Rd=10 Ω. As previously mentioned, these values were assumed for the purpose of illustration. Depending on the given application, the ranges listed in Table I may be freely adapted (see e.g. in [START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF] when focusing on resonances). The BP NGD function parameters, R, L and C were synthesized by mean of formulas ( 29), ( 30) and (31) by assuming R0=50 Ω. The POC dispersion-NGD circuit simulated in the schematic environment of the ADS® commercial tool from Keysight Technologies® is displayed in Fig. 4(a). A follower is inserted between the two circuit blocks for the inter-stage matching. The AC analysis was carried in the frequency band delimited from 0.15 GHz to 0.35 GHz. The significant flatness of magnitudes and GDs of TF, respectively displayed in Figs. 4(b) and 4(c), reveals the effectiveness of the NGD resonance reduction method. The calculated ("TF Dc ," "TF Nc " and "TF DNc ") and simulated ("TFDs", "TFNs" and "TFDNs") responses, plotted in solid and dotted curves, respectively are in excellent agreement. The dispersion, NGD and total TF responses, denoted TFD, TFN and TFDN, respectively are plotted in solid black, red dashed and blue dotted curves, respectively.



(expressed in equations ( 23) and ( 24)), under 80 MHz NGD bandwidth, an obvious reduction of resonance magnitude TFD is observed compared to TFDN, and also the GD GDTFD compared to GDTFDN in the NGD frequency band. As expected, the total TF TFDN of dispersion-NGD circuit presents a better flatness with value much closer to 0 dB (or unity TF).

B. Parametric Analyses with Respect to Rd, Ld and Cd

To demonstrate the NGD method robustness, AC analyses with respect to the dispersion variables (Rd,Ld,Cd) assumed fluctuating between -/+5% were investigated. For this parametric analyses, five different test cases were considered as seen in Table II. The resonance frequency decreases from 264 to 239 MHz. The suitable parameters (R,L,C) of NGD circuit were calculated in function of variables (Rd,Ld,Cd) via formulas (29), (30) and (31) with R0=50 Ω and GD(f0)=-10 ns. As results, we obtain the performances of the dispersion-NGD corrected circuit indicated in Table II. The dispersion, NGD and dispersion-NGD circuit responses are plotted in dashed, dotted and solid curves shown in Figs. 5. As expected, excellent corrections of resonance dispersion effect are performed for each cases of variables (Rd,Ld,Cd). Total TF close to unity is realized with the peaks of magnitude reduced from 18.5 dB to 2 dB and GD from 10 ns to 1.83 ns.

C. Parametric Analyses with Individual Variations of Dispersion Circuit Parameters, Rd, Ld and Cd

The effectiveness of the developed NGD method with the individual variations of Rd, Ld and Cd is investigated in the present subsection. In difference to the study presented in the previous subsection, in this part, we assume that the same NGD circuit with the initial parameters calculated in Table I. Then, the AC analyses of dispersion, NGD and dispersion-NGD circuits were performed. The results of each dispersion parameter effects are highlighted by the magnitude and GD responses given in Figs. [START_REF] Bertocco | Analysis and Mitigation of EMC Effects of Electric Resonances in Circuits[END_REF]. For the purpose of the illustration, it is to be noted the parametric analysis is achieved varying each of the three parasitic terms Rd, Ld and Cd, with respect with the constitutive relations ( 14) and [START_REF] Wang | A Negative Group Delay Microwave Circuit Based on Signal Interference Techniques[END_REF].

Moreover, the NGD method performances with the variations II.

IV. CONCLUSION

An original reduction of resonance dispersion effect by using BP NGD function is initiated. The mechanism of the outstanding simple and efficient NGD method is described with TF approach. The design equations of appropriated BP NGD circuit in function of the dispersion parameters are formulated. The NGD method feasibility is illustrated with frequency domain analyses by considering passive RLC circuits. Calculated and simulated results are in excellent agreement; the outstanding TF flatness improvement confirms the effectiveness of the NGD method. Parametric analyses in function of the resonance dispersion variables are carried out to illustrate the method robustness.

Applicative study with demonstrators is under the prospect of this study. As ongoing research, an application of the BP NGD correction to wireless multipath channel dispersion is currently in progress.

  )The dispersion effect associated to the resonance is also characterized by the magnitude:

Fig. 1 .

 1 Fig. 1. Dispersion (a) circuit, and TF (b) magnitude and (c) GD specifications.

  0

Fig. 2 .

 2 Fig. 2. BP NGD (a) cell and TF (b) magnitude, and (c) GD specified responses.

Fig. 3 .

 3 Fig. 3. Desired TF (a) magnitude and (b) GD specifications of (c) cascaded dispersion-NGD circuit.

Fig. 4 .

 4 Fig. 4. (a) POC circuit. Comparisons of calculated (solid line) and simulated (dotted line) TF (b) magnitude and (c) GD responses.

Fig. 5 .

 5 Fig. 5. TF (a) magnitudes and (b) GDs of dispersion (TFD), NGD (TFN) and both combined (TFDN) circuits for the different cases indicated in TableII.

Fig. 6 .

 6 Fig. 6. TF magnitudes and GDs of dispersion (dashed lines) and both NGDdispersion circuits (plain lines) for the different cases indicated in Table III under (a-b) R d , (c-d) L d and (e-f) C d variations. of Rd, Ld and Cd can be quantified with the magnitudes and GDs summarized in Tables III, IV and V, respectively. The magnitude and GD responses are shown in: • Figs. 6(a) and 6(b) for the Rd -/+5% variation by fixing Ld and Cd. It can be seen that this resistive variation does not

TABLE I POC

 I DISPERSION AND NGD CIRCUIT PARAMETERS, AND SPECIFICATIONS

			Parameter	f0	T0	τ0
	Dispersion	Value		0.25 GHz	18 dB	10 ns
	Specification	Parameter	Rd	Ld	Cd
			Value		10 Ω	50 nH	8 pF
		Parameter		Rd	Ld	Cd
	NGD	Value Parameter	Tn(f0)	10 Ω τ0	50 nH f1	8 pF f2
		Value	-17.95 dB	-10 ns	213 MHz	293 MHz
	Dispersion-	Parameters		T(f0)	GD(f0)
	NGD		Value	1.91 dB	1.68 ns

  Table I summarizes the magnitude and GD values around the considered center frequency, f0. Thanks to the BP NGD function with cut-off frequencies, 1,2

	=	1,2 2 f

  However, the NGD influence is obviously confirmed by the solid curves with reduction of magnitude and GD peaks around f0. From 18.4 dB to 1.93 dB, and 10.5 ns to 1.7 ns.• Figs. 6(c) and 6(d) for the Ld-/+5% variation by fixing Rd and Cd. As expected, it is to be noted that this inductive effect variation affects principally the resonance frequency and the peaks of both magnitude and GD. As illustrated by Fig.6(c), the NGD influence is confirmed by the decrease of the magnitude from 18.16 dB to 2.72 dB. Then, the GD is decreased from 10.51 ns to 3.67 ns. The degraded quality of NGD-dispersion responses is due to the mismatch between the dispersion resonance frequency f0 and the BP NGD centre frequency. • Figs. 6(e) and 6(f) highlight the influence of the Cd-/+5% variation by fixing Rd and Ld. In this configuration, both the resonance frequency and magnitude are changing significantly. Then, the BP NGD function reduce again the resonance effect with reduction from 17.74 dB to 2.63 dB, and the GD from 10 ns to 3.47 ns. Similar to the previous configuration, the mismatch between the dispersion resonance and NGD centre frequency explains the less significant effectiveness of the method.

variation by fixing Ld and Cd. It can be seen that this resistive variation does not affect significantly the dispersion responses.

Date of publication xxx xx, 2020. This research work was supported in part by NSFC under Grant 61601233 and Grant 61971230, in part by the Electrostatic Research Foundation of Liu Shanghe Academicians and Experts Workstation, Beijing Orient Institute of Measurement and Test under Grant BOIMTLSHJD20181003, , in part by the Jiangsu Innovation and Enterprise Group Talents Plan 2015 under Grant SRCB201526, and in part by The Priority Academic Program Development of Jiangsu Higher Education Institutions.