Comptes Rendus Physique

. Thispaperintroducesadesignmethodofsimplebandpass(BP)negativegroupdelay(NGD)topology. The fundamental speciﬁcations of BP NGD function are deﬁned. The NGD passive topology consists of parallel resistance associated with an open-ended microstrip stub. The NGD properties and characterization with respect to the constituting stub parameters are established. The validations are performed with theoretical calculated and simulated GD, transmission and reﬂection coe ﬃ cients. The BP NGD circuit can be useful for the improvement of phase linearity and GD equalization of future 5G microwave devices.


ted as a prom
sing solution for future mobile communication [3].However, the communication system using 5G wireless sensor networks (WSNs) may suffer from electromagnetic compatibility (EMC) issues, for example, due to the radio link signal interferences [4,5], EMC and massive Internet of Things (IoT) [6].

Different challenging RF and microwave device designs have been made to be under the race for the future 5G infrastructure deployment.For example, to improve the performances of the transmitter (Tx) and receiver (Rx) circuits, a reconfigurable phased array was introduced for controlling 5G terminals [7].High directive beam steering of adaptive antenna array plays an important role in the performance of such communication terminals [8].

An innovative solution of phased array design, using the unfamiliar negative group delay (NGD) function, enabling beam squint elimination was proposed [9][10][11][12].Such a phased array design method is fundamentally based on the use of NGD based non-Foster elements [13][14][15].The NGD non-Foster networks were initially designed with lossy circuits which limit the r fields of potential applications [13,14].Then, active circuit based non-Foster elements were also designed [15].In addition to the phased array networks, the NGD circuits were also used for the design of oscillators, filters [16] and phase shifter [17] for RF and microwave communication systems.Moreover, the NGD function was also used for the design of diverse devices as an absorptive band-stop filter [18] and equalization of bandpass f lters [19].One of the most specific applications of NGD function concerns the signal delay compensation [20] and propagation delay synchronisation [21].This later application is implemented with the reduction of the propagation delay based on the NGD function [21].Despite these tentative applications, so far, the familiarity of non-specialist RF and microwave engineers to the NGD function necessitates further research works.For this reason, the present paper is focused on the design method of bandpass NGD circuit for 5G frequency band.Before the elaboration of the circuit design, it is worth noticing a brief description of microwave NGD state of the art.

The NGD microwave existence was one of the curious debates between research design engineers in the 2000s.Different feasibility of NGD effect generations [22][23][24][25][26][27][28][29][30][31][32][33] was investigated.The NGD effect was experimented with signal interference techniques [22].Then, the NGD function was also designed with a microwave transversal filter approach [23].The democratization of the NGD function was also be proven with its different aspects as the possibility of dual-band design [24].The most promising NGD applications were encouraged by the number of research works on the compact and miniaturized microstrip transmission line (TL) circuit designs per-  formed in the last decade [25][26][27][28][29][30].

Because of the unfamiliarity of RF/microwave design engineers and also research co munities to this fascinating function, it would be necessary to highlight the NGD function meaning.In this way, the analogy between the filter and NGD functions was initiated in [31,32].The concept of bandpass (BP) NGD function was introduced as a function of the sign of the transmission coefficient group delay (GD) [32].Because of the open-stub based NGD topology simplicity, a design of a BP NGD cell with 5G-centre frequency was recently proposed during URSI France scientific days 2020 [33].This paper is an extension of the research work published in [33] with further detailed description of the BP NGD specifications, theoretical design and potential applications.


Preliminary definition of bandpass NGD function

A preliminary def nition of the unfamiliar BP NGD function is introduced in this section.After the theoretical expression recall, the specification parameters will be defined.The key parameters required for the NGD study will be pedagogically explored.


S-matrix black box

First and foremost, acting as a two-port system, the NGD circuit investigated in the present paper can be generalized by the black box system represented in Figure 2. By denoting the complex angular frequency variable, s = jω, this system can be analytically modelled by the 2-D S-matrix:
[S NGD (jω)] = S 11 (jω) S 12 (jω) S 21 (jω) S 22 (jω) . (1)
In contrast to the classical RF and microwave circuits, the NGD study requires more intensive analysis of frequency responses of GD calculated from the S-

trix introduced in (1).


Magnitudes of S-parame
ers

Before the NGD analytical definition, it is worth noting that the present paper is essentially focused on passive circuits.Therefore, the performance assessment of BP NGD circuit must include:

• The insertion loss or gain related to the transmission

oefficient magnitud
which should be lower than an expected value, a < 1:
S 21 (ω) = |S 21 (jω)| ≥ a (2)
C. R. Physique -2021, 22, n S1, 53-71

• The input and output reflection losses related to the reflection coefficient magnitudes which must be lower than an expected matching value, b < 1:
S 11 (ω) = |S on GD definition

By its definition, the NGD study is fundamentally dependent on the GD analytical expression.For this reason, it would be important to recall the mathematical defi

tion of this key parameter.
t is derived from the phase of the transmission coefficient of S-matrix introduced in (1).In other words, it is given by:
ϕ(ω) = arg[S 21 (jω)].(5)
The analytical model of the associated GD is d fined by:
GD(ω) = −∂ϕ(ω) ∂ω .(6)

Graphical representation of BP NGD specifications

The unfamiliar BP NGD function was inspired by ever, it is important to note that the NGD function depends essentially on the sign of GD expressed in (6) but not the behaviour of the S 21 magnitude

) = 1 − S 21 (ω 0 ). (35
Furthermore, the GD expressed in ( 26) is simplif ) = −R 0 Z τ 0 R(R 0 + 2R) . (36)

NGD design synt stub topology in function of given specifications

Simi e specifications of the BP NGD function introduced earlier circuit (see Figure 4).


Equation of TL delay

The d sired NGD centre frequency, f 0 , is the main parameter to determine the propagation delay of TL constituting our BP NGD circuit introduced in Figure 4.The analytical equation om (32).Therefore, we have the synthesis equation:
τ 0 = 0.25 f 0 . (37)

Equation of R

To determine the resistor, R, of topology introduced in Figure 4, we need the value of return loss, a < 1.Generally, to fulfil the matching expectation, a = − ned by solving the equation:
S 11 ( f 0 ) = a. (38)
In this case, as predicted by (35), the transmission coefficient i − a. (39)
Emphatically, we can establish the synthesis e

ation of TL characteristic impedance

The design equation of the TL characteristic impedance, Z , can be est
blished from the given propagation delay, which needs to be reduced from BP NGD circuit.We can denote for example this delay by the real positive, t > 0. The resolution of the following equation enables to d

n application
rinciple

This section is focused on the validation of the previous BP NGD theory.Doing this, a proof of concept (POC) designed with microstrip circuit is introduced.The ideal circuit synthesis from previously established design formulas will be introduced.Then, the comparison between the ideal model and ation is discussed.


POC description

As POC, microstrip circuits were designed and e concretely the previously developed BP NGD theory.The

he microstrip structure is illustrated i
Figure 5(a).The dielectric substrate is supposed characterized by thickness, h and the Copper-metallized conductor supposed is characterized by physical width w, length d and thickness x.The specifications of the Kapton substrate and metallization of the NGD circuit POC are indicated in Table 1.


Discussion on the obtained re n explores the validation results of the open-ended stub-based BP NGD theory.It is noteworthy that the analytical Matlab calculations of the BP NGD

ith:

• Reflection coefficient given in (22),

• Transmission coeffi
ient carried out from ( 23),

• And GD from (25).

To validate the analytical calculated results, the POC circuit introduced in Figure 5(b) was simulated with the commercial tool ADS ® from Keysight Technologies ® schematic environment.2 addresses the desired NGD specifications to design the 5G NGD POC with 3.6 GHz centre frequency.By using

esign (37), ( 40
and (42), we obtain the NGD parameters summarized in Table 3.

The present computational study was carried out in the bandwidth from f min = 3.3 GHz to f max = 3.9 GHz under more than 200 samples.

Parametric analyses were performed in function of resistor, R, and the TL characteristic impedance, Z .Figures 6(a), 6(b) and 6(c) display the cartographies of GD, S 11 and S 21 , respectively, when R is varied from 50 Ω to 100 Ω by keeping the other paramet r values the sa e as in Table 3.It can be seen that the NGD center frequency is insensitive to R. However, the GD optimal absolute value and transmission coefficient optimal value at f 0 = 3.6 GHz are d creasing when R increases.The results of the other parametric analysis versus TL characteristic impedance, Z ,

re shown in Figure 7. Figures 7(a),
(b) and (c) display the cartographies of GD, S 11 and S 21 when Z is varied from 0.5 kΩ to 1.5 kΩ by keeping the other parameters same as in Table 3. Once again, the NGD center frequency is insensitive t Z .The GD optimal absolute value and tr nsmission coefficient optimal value at f 0 = 3.6 GHz behave inversely o the case of resistor variation.

After calculations and slight optimization of TL width and length, the NGD POC operating at 3.6 GHz was designed with the parameters indicated in Table 4.

Figure 8 display the compared results from calculations ("Calc.")and ADS ® simulations ("Simu.").As expected theoretically, the BP NGD function is observed as depicted in Figure 8(a): the NG centre frequency is of about f 0 = 3.6 GHz.Table 5 summarizes the differences between the Matlab calculated and ADS ® simulated resul s.Figure 8(b) presents a comparison between the transmission coefficients.It can be seen that the loss is higher around the NGD centre frequency.Nevertheless, as depicted in Figure 8(c), the circuit is well-matched, i.e.S11 better than −10 dB over the frequency band.The slight deviation between the calculation and simulation  herein is due to the skin effect and the substrate diffraction induced in the numerical simulation.Those effects are not included in the ideal model of our circuit in the S-matrix of (16).


5.25-GHz BP NGD POC

Table 6 summarizes the desired NGD specifications to design the 5G NGD POC to operate with 5.25 GHz centre frequency with respect to the US UNII standard [34].The calculated NGD parameters by using design (37), ( 40) and (42) are given in Table 6.In an analogue way as in the previous case, the 5.25 GHz BP NGD POC was designed after calculations and slight optimization of TL width and length.The circuit was designed with he same parameters as indicated in Table 4 except the TL physical length, d = 16 cm.

Figure 11 illustrate the BP NGD behavior validation through theoretical and simulation results.Figure 11(a) presents a comparison between the calculated and simulated GDs.As expected, the NGD centre frequency is equal to f 0 = 5.25 GHz.Table 7 depicts the differences between the Matlab calculated and ADS ® simulated results.Moreover, a good correlation between the calculated and simulated transmission and reflection coefficients are observed in Figures 11(b


Discussion on 5G applications

The feasibility of the unfamiliar BP NGD function previously confirmed by the POC example raises questions on the ongoing research.The developed BP NGD concept led to think on a potential application for future design of RF and microwave devices.


Tx-Rx architecture

Among the different possibilities of application, we expect that the BP NGD responses can be used for the delay correction of multi-channel systems as the case of 802.11 a/b standards.One concrete scenario of application is illustrated by the network architecture of 5G displayed  in Figure 12.Based on this scenario, we can imagine that multiple data can be transmitted from wireless communication standards as WiFi, GS

re current cases of
ultiple connected objects, information can be transmitted from WSNs included in a Tx-Rx system.An example of WSN scenario and NGD delay effect reduction [20] between different channels, with centre frequencies-bandwidths, ( f m=1,2,3 , ∆ f m ), is illustrated in Figure 13(a).In the propagation environment with wave signal propagating at speed of light, c = 300,000 km/s, we can estimate the delay of the signal propagating from Tx(WS m=1,2,3 ) point, M m , to Rx point, M , by: In certain case of configuration as illustrated by Figure 9(a), we may consider delay differences:
t m ( f m ) = d m c .(43)       ∆t 2 = t 2 ( f 2 ) − t 1 ( f 1 ) = d 2 − d 1 c ∆t 3 = t 3 ( f 3 ) − t 1 ( f 1 ) = d 3 − d 1 c . (44)
Based on the innovative BP NGD function,

networks can be implemented w
th the configuration of Figure 13(b) [21].It consists inserting a BP NGD function in the Rx system with, for example, dual   band NGD aspect defined by:
       GD( f 2 ) = d 1 − d 2 c GD( f 3 ) = d 1 − d 3 c .(45)
In this case, assuming idea

matching between th
Rx antenna, NGD circuit and input of the Rx system, the total delay for each channel can be expected as:
               delay total ( f 1 ) = d 1 c delay total ( f 2 ) = t 2 + GD( f 2 ) = d 1 c
delay total ( f 3 ) = t 3 + GD( f 3 ) = d 1 c .

(46) The validity of the BP NGD theory is verified with a POC of microstrip circuit operating around the 5G frequencies 3.6 GHz and 5.25 GHz.The promising S-parameters and GD r

ults were obtained and discussed.As expected, results showing a BP NG
behaviour with a good correlation between calculation and commercial tool simulation is obtained.The obtained results are promising as an upcoming application of the BP NGD function for future developments related to 5G system delay correction.


Conclusion

Figure 1 .
1
Figure 1.Architecture and scenario of 5G infrastructure proposed by 5G IA [1] (Permission granted by 5G IA).


Figure 2 .
2
Figure S 11 and (c) S 21 .


Figure 4
4
Figure4represents the TL based topology of NGD passive cell connected between node M and ground node[16].This topology is assumed connected to terminal reference impedances, R 0 = 50 Ω connected between nodes M 1 M and MM 2 .The circuit can be represented as a twoport system fed by voltage sources U 1 (left) and U 2 quivalent impedance, [Z NGD (jω)], is linked to the external voltage and current vectors by the matrix relation:


CFigure 5 .
5
Figure 5. (a) Microstrip d