Novel Tee-Shaped Topology Theory of Low- and High-Pass NGD Double-Type Function

Recent investigation reports that certain electronic circuits operate with the unfamiliar negative group delay (NGD) function. It is fundamentally stated that the NGD circuits can be classified in different types. For example, we have the simplest ones as LP and HP NGD circuit types. So far, all the NGD studies are focused on simple type function. In step up of the research about the NGD electronic circuit engineering, this paper develops an original theory of electronic circuit topology operating with double NGD-type behavior. The study consists in the theorization of three-port circuit simultaneously generating LP and HP NGD function types. The proposed three-port innovative circuit under study is innovatively composed of resistive-capacitive (RC) network of Tee-shaped topology. Moreover, the LP and HP double-NGD circuit is a first order cell which does not contain any inductive component. The LP and HP NGD analyses of the Tee-topology is based on the 3-D S-matrix modelling. Analytical investigation based on classical circuit theory is elaborated to determine the S-matrix model from the equivalent admittance matrix. The identification of the double-NGD function is established from the LP- and HP-NGD canonical forms. Hence, the specific double-NGD characteristics are defined in function of the R and C elements constituting the three-port topology. To validate the developed double-NGD theory, a proof-of-concept (PoC) of SMD lumped component-based three-port circuit is designed, simulated, fabricated and tested. The calculated, simulated and measured results from the three-port circuit PoC prototype in very good agreement confirm the double-NGD behavior. The proposed innovative NGD three-port circuit design is useful in the future for the synchronization of signals propagating through multi-way communication system.

VOLUME XX, 2022 1 To meet the users demand, the future of integrated circuit (IC) and printed circuit board (PCB) design trends to be challenging in terms of integration density [1][2]. An adequate synthesis approach is needed to run the PCB routing notably for high density (HD) and high speed (HS) circuit [3][4]. An innovative design method must be especially developed the digital PCB tree clock. Therefore, optimal methods of modelling [5] and simulation [6] are necessary. For example, innovative methods of clock signal distributions were proposed to ensure the electronic system synchronization [7]. A novel methodology for optimizing the global interconnect width and spacing of technology nodes was reported by International Technology Roadmap for Semiconductors (ITRS) [8].
An alternative tentative solution to equalize the group or propagation delay effects was also initiated by using negative group delay (NGD) function [33][34]. However, the NGD interconnect effect equalization is not welldeveloped because of the non-specialist electronic design engineer misunderstanding on the NGD function.
For this reason, further didactical research work on the NGD function must be forwarded. Design methods of NGD active circuits with amplifier associated to resistiveinductive-capacitive (RLC) network were so far developed [35][36][37][38]. It was stated that the NGD effect signature in the time-domain corresponds to the possibility to propagate the output signals in time-advance compared to the corresponding inputs [35][36]. Because of such a counterintuitive effect, the interpretation of the NGD function intrigues most of electronic design and research engineers. Therefore, an easier to understand NGD theory inspired by the analogy with the filter function was initiated [39]. The similitude between the NGD and filter was established thanks to the regular conformity between the group delay (GD) and the magnitude of linear circuit following the Kramer-Koenig theorem. Certain lumped NGD circuits can be classified as low-pass (LP) [39][40][41] and high-pass (HP) [39,[42][43][44] NGD topologies. The NGD circuits [33][34][35][36][37][38][39][40][41][42] cited herein were constructed by using lumped R, L and C components. Moreover, in difference to the classical circuit theory as previously explored for the case of interconnect structures , the NGD circuits [33][34][35][36][37][38][39][40][41][42][43][44] available in the literature were designed as: • Two-port topologies of passive or active circuits, • Using RL, RC or RLC networks, • Or using resonant networks.
Therefore, the existence of double type NGD topologies constituted by three-port circuits is the main focus of the present paper. The developed NGD theory is based on the original Tee-shaped passive topology of resistive-capacitive network. The paper is divided in six main sections as follows: • Section II states the definitions of fundamental parameters allowing to represent the LP-and HP-NGD responses. For the basic understanding, the representation is using the case of two-port circuit. • The simplest canonical forms of first order transfer functions representing LP-and HP-NGD circuit types are investigated in Section III. • Section IV focuses on the analytical calculation of the S-matrix model representing the Tee-shaped topology. • Section V describes the LP and HP double-NGD theorization of the Tee-shaped topology of RCnetwork. The main characteristics of the double-NGD topology are established in function of the capacitor parameters constituting the circuit. • Section VI discusses the validation results by designing a three-port circuit which is a PoC of double-NGD prototype. • Section VII finalizes the paper with the conclusion.

II. DEFINITION OF LP-AND HP-NGD CIRCUIT TYPE PARAMETERS
The present section addresses the main parameters necessary for the NGD analysis. Acting as an unfamiliar function compared to the classical ones (filter, antenna, amplifier, phase shifter, oscillator, …), it is necessary to recall the LP-and HP-NGD specifications.

A. BASIC REPRESENTATION PARAMETERS FOR NGD ANALYSIS
The present section addresses the main parameters needed for the NGD analysis. Acting as an unfamiliar function, it is necessary to recall the LP-and HP-NGD specifications.
By taking sj =  as the Laplace variable in function of the angular frequency ω, for the case of symmetrical circuit, the S-matrix model generally shown by the black box of Fig. 1 can be expressed as: (1) The main parameters to be analyzed in this circuit are: • The magnitudes of reflection and transmission coefficients, respectively: 11 11 21 21 ( ) ( ) The NGD response of circuit represented by S-matrix of Fig. 1 is defined by the existence of frequency band where the GD expressed in equation (3) becomes negative: (6) Similar to the filter behavior, we can distinguish different types of NGD responses related to the frequency band where condition (5) is satisfied. The NGD cut-off angular frequency, 00 2 f =  , which should be unique for the case of first order circuit [39][40][41][42][43][44] is defined as the root of equation: Based on these definitions, we can represent the LP-and HP-NGD function specifications.

B. LP-NGD SPECIFICATIONS
The LP-NGD ideal specifications can be represented the GD spectral diagram shown in top of Fig. 2(a). Similar to the filter electronic function, in this case of LP-NGD type, the first parameter NGD bandwidth is equal to the cut-off frequency 00 2 f =  . We can point out that condition (5) is satisfied when 0  . The NGD frequency band should not reach the shadowed part of the spectrum where the angular frequency 0   . The second parameter of LP-NGD type is the NGD value. It can be defined by:

C. HP-NGD SPECIFICATIONS
Similar to the filter behavior, the HP-NGD type has an NGD frequency band in the opposite case of LP-NGD one defined in the previous subsection. The HP-NGD type ideal responses can be represented by Figs. 3. In this case, the GD spectral diagram is shown by Fig. 3(a). VOLUME XX, 2022 1 Therefore, some specifications of the reflection and transmission coefficient responses will be proposed in the following subsection.

D. S-PARAMETER SPECIFICATIONS IN THE NGD FREQUENCY BAND
In addition to the GD specifications, the S-parameter magnitudes should satisfy some criteria in the NGD frequency bands. The reflection coefficient ideal responses of LP-and HP-NGD types are represented by Fig. 2(b) and Fig. 3(b), respectively. The reflection coefficient is specified by the given maximal value of real positive constant, A<1 or AdB<0. In the NGD frequency bands ( 0   for the LP-NGD type and 0   for the HP-NGD type), these magnitude diagrams are assumed to be plotted in dB. In this study, the reflection coefficients can be defined by: (11) As a more concrete approach, the canonical forms of LPand HP-NGD circuit types are explored in the following section.

III. INVESTIGATION ON THE GENERAL CANONICAL FORMS OF 1 ST ORDER LP-AND HP-NGD TRANSFER FUNCTION TYPES
The most practical way to study the NGD circuit is the consideration of canonical transfer function types. The present section is focused on the main parameters and properties of LP-and HP-NGD types with first order transfer function.

A. BASIC PARAMETERS OF THE TRANSMITTANCE CANONICAL FORMS
The simplest canonical transfer function forms of LP-and HP-NGD circuit types can be represented by first order system [39][40][41][42][43][44]. To write the fundamental transfer function associated to the LP-NGD and HP-NGD circuit types, we should consider the specification parameters represented by diagrams previously shown by Fig. 2(a) and Fig. 3(a), respectively. These canonical forms of transfer function can be expressed knowing the NGD specifications as: • the cut-off frequency, 0  which is defined by equation (7), • and also, the GD value at very low frequency defined by: ( 0) n GD =  (12) By using these parameters, the most important canonical forms of NGD circuits will be expressed in the following subsection.

B. GENERAL TRANSFER FUNCTION ASSOCIATING LP-AND HP-NGD CIRCUITS
Once again, by inspiring from the LP-and HP-filter theory, the transfer functions of all LP-and HP-NGD circuits can be formulated by first order system. The main and basic transfer function can be identified by the expression written as:

T a s Ts bs
with: According to the NGD theory [39][40][41][42][43][44], the GD at very low frequencies defined in equation (12) is given by: Furthermore, the NGD cut-off angular frequency is given by: Now, we may wonder on how to identify if our canonical transfer function represents LP-or HP-NGD circuit types.

C. CASE OF LP-NGD CIRCUIT TYPE
More and more theoretical and practical validation studies As explored in the next subsection, the HP-NGD circuit type [42][43][44] is much more difficult to express.

D. CASE OF HP-NGD CIRCUIT TYPE
In the opposite case of The optimal GD defined by equation (20) is given by: The transmission coefficient magnitude ( ) ( ) T T j =  at the optimal frequency will be: (24) As application of the canonical transfer function forms, we need the expression of transmission coefficient of the circuit under study. The next section will develop the necessary S-matrix model.

IV. S-MATRIX MODEL OF THE TEE-SHAPED TOPOLOGY
The fundamental theory of lumped circuit is elaborated in the present section. By means of Tee-shaped topology, the present NGD theory is built with 3-D S-matrix analytical model. Before the analytical exploration, the proposed topology will be described. Then, based on the Kirchhoff circuit laws (KCL), the admittance equivalent model will be established before the S-matrix extraction.

A. TOPOLOGICAL INTRODUCTION
The general configuration of the proposed Tee-shaped topology under study is introduced in Fig. 4. It acts as a three-port circuit fed by voltage sources, Vk=1,2,3 through access node Mk. The middle node is denoted by M. Each branch MkM presenting a series impedance, Zk, is traversed by current, Ik with the subscript k=1,2,3. The fundamental and equivalent model of this proposed topology can be elaborated from the generalized Ohm's law expressed as: The associated admittance matrix is analytically represented by: To determine the constituting elements of this admittance matrix, we propose the theoretical investigation in the following paragraph.

B. ANALYTICAL EQUATIONS FOR EXTRACTING THE EQUIVALENT ADMITTANCE MATRIX
The proposed analytical expression of our proposed topology depicted by Fig. 4 can be established by the basic circuit theory. In this way, the application of the node Kirchhoff circuit law (KCL) at node M gives the equation: (27) The KVL along the mesh GND-M1MM2-GND enables to write: Under the similar way, the same law applied to mesh GND-M3MM2-GND gives: These basic equations serve to the development of the Teeshaped topology S-matrix equivalent model which will be obtained via the admittance matrix in the following subsection.

C. GENERAL EXPRESSION OF THE PROPOSED TOPOLOGY S-MATRIX FROM THE ADMITTANCE MATRIX
The expressions of the proposed topology branch current can be determined in function of the voltage sources and the branch impedances by solving the equation system constituted by relations (27), (28) and (29). The matrix representation of the rewritten solution lead to the Ohm's law of equation (26) with the admittance matrix: According the Y-to-S transform, the S-matrix can be determined from the admittance one expressed in equation (30): with: • the 3-D identity matrix: (33) • and the reference impedance R0=50 Ω. Accordingly, we have the proposed topology following transmission coefficients in function the branch impedance:

R R Z s S s S s
Ds + == (36) with the denominator quantity: The associated reflection coefficients can be written as: This general expression will be established to determine the RC network-based three-port circuit. Thus, the expressions enable to explore the NGD analysis in the next section.

V. NGD ANALYSIS OF THE THREE-PORT TEE-CIRCUIT
This section introduces the theoretical concept of the threeport Tee-shaped circuit NGD-type identification. The basic approach allowing of the NGD analysis will be developed from the TF canonical forms explored in paragraph II-E.

A. S-MATRIX MODEL OF RC NETWORK-BASED THREE-PORT TOPOLOGY
Fig . 5 represents the corresponding Tee-shaped topology by using RC-network. This concrete three-port circuit is designed with RC-parallel network connected to port 1 to the middle point. The analytical expression can be obtained substituting the impedance by: VOLUME XX, 2022 1 Consequently, the reflection coefficients expressed in equations (34), (35) and (36) become respectively:

RCs S s S s Ds
The corresponding denominator written in equation (37) becomes: Thus, the reflection coefficients expressed in equations (38), (39) and (40) becomes: The NGD theorization of our lumped Tee-topology will be defined from this S-matrix model by means of the NGD canonical forms.

B. NGD ANALYSIS OF PORT-PORT TRANSMISSION
We can establish from the transmission coefficient expressed in equation (42) the NGD analysis corresponding to transmission through port-port. By identification with the TF expressed in equation (13), we have the following coefficients: We can demonstrate that the GD at very LF will be: We can remark that this GD is always negative whatever the values of R and C. Then, the cut-off frequency should be: These last two formulas confirm that our three-port Teetopology behaves as a LP-NGD type for the transmission through port-port.

C. NGD ANALYSIS OF PORT-PORT TRANSMISSION
In this case, the analysis can be performed by considering the transmission coefficient expressed in equation (43). This corresponds to the NGD analysis associated to the transmission through port-port. By identification with the TF expressed in equation (13), we have the following coefficients: We can demonstrate that the GD at very LF will be: We can remark that this GD is unconditionally positive whatever the values of R and C. Then, the cut-off frequency should be: These last two formulas confirm that our three-port Teetopology behaves as a HP-NGD type for the transmission through port-port. The optimal frequency expressed in equation (22) The minimal GD formulated in equation (23) is transformed as: Then, the attenuation at optimal frequency given in equation (24)

D. SYNTHESIS EQUATION FROM PORT-PORT LP-NGD SPECIFICATIONS
The methodology of LP-NGD type synthesis equations are elaborated in this subsection. The following synthesis equation is based on the desired values of: • Attenuation, A<1 • And the NGD value, 0 0  

1) RESISTOR SYNTHESIS EQUATION
By inverting equation (47), we have the synthesis equation of resistor: (60) It means that to get a realistic value of resistor, we must choose: By considering equation (60), Fig. 6 plots the variation of resistor R versus the attenuation A varied from -20 dB to -3.522 dB. We find that the resistor is decreasing from 0 to 425 Ω.

2) CAPACITOR SYNTHESIS EQUATION
By inverting equation (50), we have the capacitor formula: Substituting the resistor formula of equation (60) into previous one, we can also determine the capacitor by the equation: Fig. 7(a) represents the linear plot of ratio capacitor by absolute value of NGD value for attenuation A varied from -20 dB to -3.522 dB. For the better understanding about the variation from 0.0028 F/s to 132 F/s, the y-scale semilogarithmic plot is proposed in Fig. 7(b). The previously established synthesis formulas will also affect the other NGD parameters corresponding both port-port to port-port and transmissions.

E. EFFECT OF SYNTHESIS FORMULAS ON PORT-PORT PARAMETERS
The LP-NGD synthesis formulas affect systematically the LP-NGD cut-off frequency and also the reflection coefficients. The analytical investigation on these impacts of chosen desired parameter as attenuation A will be studied in the following paragraphs.

1) LP-NGD CUT-OFF FREQUENCY
Substituting the previously established formulas of R and C into the cut-off frequency written in equation (51), we can demonstrate that we have: It yields from this equation the plot of product 021 0  shown by Fig. 8 for attenuation A varied from -20 dB to -3.522 dB. We can remark that the product is decreasing quite linearly when the attenuation is increasing.  (45) and in equation (46), become, respectively: The HP-NGD aspect parameters of transmission from port-port will also be influenced by the synthesis equations.

F. EFFECT OF SYNTHESIS FORMULAS ON PORT -PORT PARAMETERS
The LP-NGD synthesis formulas influence also the specifications of the HP-NGD behavior. The following paragraphs shows the expressions and ranges of variation of HP-NGD parameters as attenuation, frequencies and optimal GD versus chosen attenuation A varied from -20 dB to -3.522 dB.

1) HP-NGD ATTENUATION
Substituting the previously established synthesis formulas of equation (60) and equation (63) into the cut-off frequency written in equation (52), we have the transmission coefficient: In this case, the HP-NGD function attenuation at very low frequency is decreasing from about -0.3 to -.5 dB as seen in Fig. 9.

2) HP-NGD CHARACTERISTIC FREQUENCIES
Following the same approach, the cut-off frequency of equation (56) is simplified as: Similarly, the optimal frequency given in equation (57) becomes: The corresponding abacuses are represented by black and red lines of Fig. 10, respectively. We can remark that despite the complexity of equations (68) and (69), both frequencies are decreasing until reaching zero value if we increase attenuation A.  (72)

3) HP-NGD OPTIMAL NGD AND ASSOCIATED ATTENUATION
In this case, we proposed to plot the ratio: (73) Fig. 11 represents the variation of the ratio versus attenuation A. We remark that the ratio is increasing with the attenuation.

FIGURE 11. Ratio of HP-NGD optimal GD and NGD value versus A.
Then, the HP-NGD optimal attenuation previously established in equation (59) will become: In this case, the variation of the HP-NGD optimal attenuation versus A is highlighted by Fig. 12. As concrete illustration of this unfamiliar double NGD theory applied to three-port circuit, the following section introduces a verification study. The obtained experimental results from PoC circuits will be examined in the following section.

VI. VERIFICATION STUDY OF THE DEVELOPED DOUBLE NGD THEORY WITH TEE-CIRCUIT POC
The feasibility study of the developed double NGD theory is described in the present section. As PoC, a prototype of synthesized, designed and fabricated Tee-shaped circuit will be presented in the next subsection. Then, the calculated, simulated and measured results will be discussed.

A. POC DESCRIPTION
The prototypes of Tee-shaped circuit were designed and fabricated. The realization was carried out by means of preliminary calculations using the synthesis equations previously established in subsection V-D.

1) CALCULATED IDEAL COMPONENTS
The Tee-shaped circuit PoCs are designed in function of the desired LP-NGD specifications A and τ0. Table I addresses the considered specification of three different cases of double NGD tee-shaped prototypes. The employed resistor and capacitor values were calculated following two synthesis equations of equation (60) and equation (63), respectively. Consequently, we get the calculated values of electrical parameters indicated in Table I. The values of corresponding LP-and HP-NGD cut-off frequencies calculated from formulas (64) and equation (68) are also indicted in Table I. To fabricate the associated circuit prototypes, we consider the resistor and capacitor nominal values belonging to E48 series. Therefore, we cannot have tolerances better than 5% of their nominal values.

2) DESIGNED AND FABRICATED CIRCUIT PROTOTYPES
The Tee-shaped circuit prototypes were fabricated by using R and C SMD components. Before the realization, the circuits were designed in the ADS® schematic environment as shown by Fig. 13(a). The calculations of theoretical model were performed with MATLAB® commercial tool. The ADS® software is an RF and microwave circuit design and simulator commercial tool. Then, the photograph of the fabricated circuit is displayed by Fig. 13(b).
The Tee-shaped circuit prototypes were implemented on Cu-metalized FR4 dielectric substrate in hybrid technology. The substrate physical characteristics are indicated in Table II.

B. SIMULATED AND MEASUREMENT RESULTS FROM S-PARAMETERS
The experimental validation results of our double NGD theory of Tee-circuit will be elaborated in the following paragraphs.

1) EXPERIMENTAL SETUP
Similar to all classical microwave circuits, the NGD circuit measurement consists in recording the touchstone model of the circuit. Our experimental solution is based on the measurement of two-port S-parameters instead of threeport one which is the natural adequate solution for our LP-HP circuit. The illustrative diagram of the experimentation is displayed in Fig. 14(a). This experimental solution is due to the only available Vector Network Analyzer (VNA) in our laboratory. Accordingly, the fabricated Tee-shaped VOLUME XX, 2022 1 circuit prototype, previously introduced in Fig. 14(b), was measured with VNA (Rohde & Schwarz ZNB 20, within the frequency band 100 kHz to 20 GHz). It is noteworthy also that during the measurement process, the Sparameters test was made under SOLT calibration. Thanks to the success of the experimental tests, we obtained the results discussed in the following paragraphs in the frequency band from fmin=100 kHz (the lowest achievable with the available VNA) and fmax=100 MHz.

2) DISCUSSION ON THE GD RESPONSE RESULTS
Comparisons between the GD responses of calculated ("Calc." plotted in black solid lines), simulated ("Simu." plotted in blue-sky dashed lines) and measured ("Meas." plotted in red dashed lines) results are discussed in the present paragraph. Indeed, the obtained results from two different key transmission coefficients are separately presented as follows: For the first case, the GDs GD21a, GD21b, and GD21c corresponding to transmission coefficients between port-port of Tee-shaped RC-network based prototype1, prototype2 and prototype3 which are defined in Table I are plotted in Fig. 15(a), Fig. 15(b) and Fig. 15(c), respectively. It can be pointed out that the three GD responses plotted in Figs. 15 behave as a typical LP-NGD function. We can see in the zoomed plots of Fig. 17(a), Fig.  17(b) and Fig. 17(c) that based on the theoretical calculation and simulation, we can assess GD21a(f≈0)=-3 ns, GD21b(f≈0)=-3 ns, and GD21c(f≈0)=-4 ns, as expected in the specifications addressed in Table I. Then, very good correlations between the calculated, simulated and measured results of LP-NGD circuit are confirmed. For the other case, GD32a, GD32b, and GD32c corresponding to transmission coefficients between port-port for the same tested prototypes are plotted in Fig. 16(a), Fig. 16(b) and Fig. 16(c), respectively. In this case, we can understand that the GD responses present rather typical HP-NGD behavior. We can see in the zoomed plots of Fig. 17(d), Fig. 17(e) and Fig. 17(f), that we have positive GDs at very low frequency. Furthermore, the GDs become negative when the frequency is higher than 18 MHz for the three tested circuit prototypes. Once again, the calculated, simulated and measured GDs are literally in good agreement.
Despite the successful verification of the double LPand HP-NGD function behavior, we can underline here that the results from measurements are slightly shifted compared to the two other ones. The observed differences are mainly due to the dispersion of the circuit substrate parameters, the circuit electrical interconnections, the measurement systematic errors and also the used lumped component fabrication tolerances.

3) DISCUSSION ON THE S-PARAMETER MAGNITUDES
In addition to the GD responses, the RF NGD circuits are expected to operate under requirements with respect to the transmission and reflection coefficients. For this reason, comparisons between the calculated, simulated and measured results of transmission coefficient magnitudes are discussed in the present section. The comparisons of transmission coefficients corresponding to port-port and port-port are plotted in Figs. 18 and Figs. 19, respectively. As expected in Table I, the transmission coefficients of Tee-shaped RCnetwork based port-port prototype1, prototype2 and prototype3 are displayed respectively in Fig. 18(a), Fig.  18(b) and Fig. 18(c).
The transmission coefficients of port-port prototype1, prototype2 and prototype3 are displayed respectively in Fig. 18(a), Fig. 18(b) and Fig. 18(c). As expected in Table I, these transmission coefficients are higher than -6 dB, -7 dB and -6 dB, respectively. For the case of port-port, we have the transmission coefficients presented in Fig. 19(a), Fig. 19(b) and Fig.  19(c), respectively. In this case, all the transmission coefficients are better than -4 dB in the considered working frequency band lower than fmax.  The reflection coefficients of the double NGD three-port Tee-shaped RC-network based circuit were also investigated. Fig. 20(a), Fig. 20(b) and Fig. 20(c) display S11 of prototype1, prototype2 and prototype3. Then, associated reflection coefficients S22≈S33 are also proposed in Fig. 21(a), Fig. 21(b) and Fig. 21(c), respectively. It can be emphasized here that all the reflection coefficients are literally at worst around -9.5 dB. This means that the double both LP-and HP-NGD topology under study is susceptible to operate with less attenuation loss and good access matching.

C. STATE-OF-THE-ART COMPARISON OF NGD SPECIFICATIONS
Despite the progress of research interest on NGD circuit design, many efforts are still needed to increase its design familiarity compared to other classical electronic circuit design.

V. CONCLUSION
An original investigation on Tee-shaped passive topology operating with three-port circuit is developed. The basic definition of parameters allowing to specify the LP-and HP-NGD function is described. The modelling of the Tee-shaped topology with 3-D S-matrix is established. The S-matrix study leads to the LP-and HP-NGD analyses. The original implementation of both double LP-and HP-NGD aspects is initiated in the first time. The existence condition of LP-NGD aspect from one branch of the Tee-circuit is proposed. Also, the existence condition of HP-NGD aspect through another branch of the Tee-circuit is expressed. The synthesis equations enabling to determine the resistor and capacitor elements are formulated.
The verification of the theoretical approach is performed with MATLAB® computation of the developed Sparameter model. Simulation was also run with commercial simulation tool. A three-port circuit comprised by SMD resistor and capacitor components is designed, implemented and tested. Then, measurements were realized in RF frequency band. As expected, the obtained results confirm outstandingly the validity of the double LPand HP-NGD aspects. Moreover, calculated model, simulations and measurements in good agreement are presented.  He is regularly involved to participate and to lead national and international research projects (ANR, H2020, FP7 KIC-InnoEnergy…). He directed and co-supervised 18 PhD and 15 Master students. His research interests involve AMS and RF circuit design from transistor to architectural level. His present research activity is focused on ultralow power circuit design for smart sensor interface and embedded electronic in bioelectronic and e-health applications, wireless systems, design technique and architecture for multi-standard transceiver. He is author or co-author of 11 patents and more than 150 papers published in refereed journals and conferences. He is an expert for the ANR, the French Agency for Research. He has served on program committees of IEEE NEWCAS and ICECS. He has been and is a reviewer of contributions submitted to several IEEE conferences and journals such as ISCAS, NEWCAS, MWSCAS, ESSCIRC, ESSDERC, RFIC, IEEE Transactions on Circuits and Systems I and II, IET Electronics Letters.