Synthesis of RF Circuits with Negative Time Delay by Using LNA

A demonstration of the negative time-delay by using active circuit topologies with negative group delay (NGD) is described in this paper. This negative time delay is realized with two different topologies operating in base band and modulated frequencies. The first NGD topology is composed of an RL-network in feedback with an RF/microwave amplifier. Knowing the characteristics of the amplifier, a synthesis method of this circuit in function of the desired NGD values and the expected time advance is established. The feasibility of this extraordinary physical effect is illustrated with frequencyand time-domain analyses. It is shown in this paper that by considering an arbitrary waveform signal, output in advance of about 7 ns is observed compared to the corresponding input. It is stated that such an effect is not in contradiction with the causality. The other NGD topology is comprised of a microwave amplifier associated with an RLC-series resonant. The theoretical approach illustrating the functioning of this NGD circuit is established by considering the amplifier Sparameters. Then, synthesis relations enabling to choose the NGD device parameters according to the desired NGD and gain values are also established. To demonstrate the relevance of the theoretic concept, a microwave device exhibiting NGD function of about -1.5 ns at around 1.19 GHz was designed and analyzed. The NGD device investigated in this paper presents advantages on its faculty to exhibit positive transmission gain, the implementation of the bias network and matching in the considered NGD frequency band.


Introduction
Recent studies revealed that certain passive [1][2][3][4][5][6][7][8] and active [9][10][11][12][13][14][15][16][17][18] circuits are susceptible to exhibit the negative group delay (NGD) function at low-and RF-/microwavefrequencies.Theoretical and experimental analyses confirm that this innovative physical function allows to generate an output signal with wave fronts propagating in advance of its input under certain conditions [7][8][9][10][11][12][13][14].Inversely to the ordinary medium, in this case, the time delay can be assumed as negative [4][5][9][10][11][12][13][14][15][16].It was pointed out [19][20] that this NGD effect does not forbid the causality principle.The limits and physical meaning of the negative time delay in function of the NGD transfer function parameters are stated in [21][22].At the beginning, the occurrence of the NGD was initially explained with the anomalous dispersion [23][24][25][26][27].The original theoretical study on the abnormal media with negative group velocity is introduced in [24].In fact, it was shown that at certain wavelengths, these media is capable to present refractive group index n g can be negative.The theoretical demonstration of the NGD can be illustrated with the following simple mathematical analysis.First of all, it is well-known that the group velocity is linked to the speed of light in the vacuum c by the expression: Clearly, if n g is negative, the group velocity v g is also negative.It means that when using a physical medium with geometrical length d, the group delay, by definition given by: can also be negative.Because of the inherent fact of the wave reshaping or the combination of the constructive and destructive interferences at the edge of the abnormal dispersive passive medium, the NGD phenomenon is systematically accompanied with significant losses, in particular, at microwave wavelengths [1][2][3][4][5][6][7][8].So, the applications of NGD passive circuits are still limited before the middle 2000s.
To overcome this limitation, active microwave topologies based on the RF/microwave transistors have been developed [14][15][16][17][18][21][22].Then, different applications of the NGD function were proposed in the literature.One of the most published applications are the design of independent frequency NGD phase shifter [28][29][30][31] and broadband active balun [32], feed forward amplifier [33][34], microwave circuit with pulse compression [35] and also the reduction of the propagation delay in the microelectronic interconnect with the NGD equalization technique [36][37][38][39][40].However, because of the transistor access-matching difficulties, the complexity of the biasing networks and the output voltage sign inversion compared to the input is found with certain NGD topologies as proposed in [14][15][16][17][18].It means that somehow the use of the transistors is more and more sophisticated.To avoid the complexity of bias network and the issues related to the matching level, the replacement of the transistor by an integrated low noise amplifier (LNA) was introduced in [41] for the base band applications.
The present paper is focused on the study of NGD topologies using microwave LNA.For the better understanding, this paper is organized in three main sections.Section 2 presents the analysis of the base band NGD topology proposed briefly in [41].This first NGD circuit is aimed to the base band applications.Section 3 is focused on the investigation of the NGD LNA topology dedicated to the modulated microwave signals application.Section 4 is the general conclusion of the paper.In order to achieve a global perspective on the frequency analysis including the stability, transfer function and matching access, the present analytical approach is based on the S-parameters analysis.To do this, an ideal microwave LNA operating in base band frequencies is considered.For the sake of the analytical expression simplification, in the remainder of this section, the considered amplifier is substituted by the ideal S-parameters expressed as:

Investigation on an NGD circuit with LNA operating in baseband frequencies
where ρ and T represent respectively the return loss and transmission parameter.Moreover, this amplifier is supposed perfectly unilateral, S A12 = 0.The theoretical analysis of this topology by using the S-parameters is introduced in the next subsection.Then, numerical demonstrations of the negative time delay will be explored based on the analyses in frequency-and time-domains.

Theory on the baseband topology proposed
After the calculation of the S-parameters of the cell shown in Figure 1, we will establish the synthesis relations enabling to determine the elements R and L in function of the desired gain and NGD values knowing the characteristics of the amplifier.

S-parameter analysis of the NGD cell for base band frequencies
As the circuit proposed is comprised of an RL-network in feedback with an amplifier, it is easier to calculate the Sparameters from the admittance matrix.According to the circuit and system theory, the admittance matrix of the whole circuit which is also the sum of the passive part admittance matrix and that of the amplifier is expressed as: where ω is the angular frequency, 1 − = j and R 0 = 50 Ω is the reference impedance of the source and the load.Through the admittance to S-matrix transform, one establishes that the following S-parameters of the NGD cell understudy are: where Therefore, the transmission phase We find that at very low frequencies ( 0 ≈ ω ), these Sparameters become: According to relation (13), the NGD cell requires the following stability condition: Moreover, it can be demonstrated from equation ( 10) that at very low frequencies ( 0 ≈ ω ), the group delay of the circuit which is analytically defined as: is expressed as: It means that to generate NGD with the circuit introduced in Figure 1 at very low frequencies ( 0 ) 0 ( < τ ), the following condition must be satisfied:

Synthesis relations
At the beginning of the synthesis process, the used amplifier is supposed characterized by its S-parameter elements T and ρ.Thus, from the desired values of gain τ , the synthesis relations enabling to determine the NGD circuit parameters R and L are extracted by inverting relations ( 15) and ( 18) respectively.From where are established the following formulations [16]: This expression is dedicated to the synthesis with ) 0 ( τ assigned as a real negative.Under the following condition: the result of expression ( 20) is physically meaningful only under the following condition: 2

Fundamental characteristics of the baseband NGD cell understudy
One of the main properties which characterizes the NGD cell understudy is based on the relation between its transmission parameter S 21 (0) and its reflection parameters S 11 (0) = S 22 (0).In fact, giving the transmission parameter T and the return loss ρ of the amplifier A, one demonstrates that the return losses S 11 (0) = S 22 (0) are mathematically linked to the insertion gain S 21 (0) by the relation: To achieve the basic stability condition: once again, the transmission parameter S 21 (0) must verify inequality (23).Furthermore, under condition (19), the group delay ) (ω τ of the NGD cell can be negative from 0 ≈ ω to the cut-off frequency c ω .Via the derivation of the transmission phase with respect to ω , it is established that this cut-off frequency is expressed as: In order to verify the relevance of the above theoretic concept, in the next section, we will examine the Sparameters and the group delay of a proof of concept device designed with the NGD topology proposed.

Numerical demonstration of the negative time effects
The simulation results presented in this section were generated from the well-known the commercial software Advanced Design system (ADS) from Agilent TM .

Design of the prototype of base band NGD device tested
As illustrated in Figure 2, a circuit composed of two-stage NGD cells was designed and simulated.After the application of the previous synthesis relations accompanied with a slight optimization of the RF-components, the choice of the relevant nominal values of passive elements were carried out.Then, comparative simulations were performed between the circuit comprised of ideal RF-part and the complete circuit shown in Figure 2.This latter was simulated with the Sparameter realistic models of passive components R, L and C available in ADS library.Table 1 indicates the references of the components employed.The employed amplifiers are MMIC LNAs HMC 395 provided by HITTITE TM .In this study, the amplifiers were biased with the DC voltage V dd = 5 V and I dd = 20 mA.
The considered MMIC LNAs were replaced by the measured S-parameters including the bias networks provided by the manufacturer.
Table 1: References of the passive components constituting the circuit shown in Figure 2.

Nominal value
Reference Manufacturer

Frequency domain investigations
One points out that the simulation results presented in this paragraph were realized by considering first the ideal RF circuit and then, the realistic circuit shown in Figure 2 by considering the touchstone or S 2 P (or two-ports Sparameter) models of the passive components provided by the manufacturer.According to the frequency bands of the components utilized, S-parameter simulations were performed.As displayed in Figure 3, a good agreement between the frequency simulations with the ideal and realistic circuits is found in the NGD frequency band.As forecasted in theory, group delay presenting negative values were realized up to about 90 MHz.This NGD value goes down to about -7 ns at very low frequencies (ω ≈ 0).As shown in Figure 4, the circuit simulated guarantees an average gain more than 0 dB from very low frequencies up to hundreds MHz.

Stability analysis
The access parameters S 11 and S 22 of the circuit tested are plotted in Figure 5.We remark that these parameters confirm the stability of the circuit from DC to 5 GHz.Moreover, in this ultra wide frequency band, the isolation parameter S 12 is better than -20 dB.To strengthen the stability of the tested device, the stability factor was also examined.One recalls that this factor is defined by: As reported in [17], this behaviour of the transmission parameter magnitude can be exploited to generate the pulse compression effect.The Rollet factor of the circuit is also plotted in Figure 6.As conclusion, the NGD topology proposed in this section respects all the constraints of classical RF active devices as the stability condition in UWB frequency band.

Time-domain investigations
For the better understanding about the meaning of the NGD effect generated by the circuit understudy, time-domain simulations were also performed in the transient option of SPICE-ADS environment.Figure 7 displays the configuration of the simulated circuit which is loaded by R 0 = 50 Ω.After ADS time-domain simulations, the results plotted in Figure 8 and Figure 9 were obtained.We can see that as explained by Figure 8, the output V out of the NGD circuit is practically conform to the Gaussian input V in which presents here a half height half width of about 18 ns.Moreover, contrarily to the ordinary circuit transient response, thanks to the NGD phenomenon, the wave fronts of the output V out is slightly in advance of about 7 ns compared to the input V in .
T im e (n s ) 0

S im u la te d
v o lta g e 0 V 0 1 0 0 5 0 In other hand, by injecting an arbitrary waveform input signal presented by V in of Figure 9, once again, an effect of the negative time-delay with time-shift of about 7 ns is found.Despite this outstanding effect, it is interesting to note that the output signal appears after the input at the starting point t = 0 s.So, the causality principle is still respected.As reported in [18], this advance phenomenon occurs only when the input spectrum belongs in the NGD frequency band.

Comments and discussions
An innovative NGD active circuit topology susceptible to operate at baseband frequencies is developed by using a microwave amplifier.Compared to the NGD topologies introduced in [14], the proposed one presents a facility to achieve a good access matching and enables to avoid the opposite sign between the output and input voltages for the single or odd number of stage circuit.Theoretical approaches based on the S-parameter analysis were proposed.Then, the synthesis relations enabling to design the electronic device with the NGD topology understudy were analytically demonstrated according to the desired values of the gain and NGD.Then, the theoretic characteristic specific to the circuit analyzed vis-à-vis the NGD effects were also established.To illustrate the relevance of the theoretic principle proposed, design and simulations of two-stage circuit based on the use MMIC LNA device were carried out.So, it demonstrates the possibility to generate NGD simultaneously with gain in baseband up to microwave frequencies.Moreover, it was stated with SPICE simulation that the presented NGD topology respects all the criteria of classical active RF circuits.Furthermore, it was shown also that the NGD circuit investigated can be employed to reduce the pure delay induced by telecommunication devices as the microwave TLs.As application, this NGD circuit is potentially useful for the delay reduction as proposed in [37][38][39][40] notably for the correction of interconnections degradation on the PCB and for the high-speed microelectronic systems as illustrated in [41][42][43].

Theory on the NGD circuit operating in RF/microwave frequencies
Similar to the previous section, to examine the functioning of the circuit described in Figure 10, we analyse its Sparameters and establish the synthesis relations enabling to determine the elements of the NGD topology in function of the desired NGD and gain values.Then, properties of the topology will be established.

S-parameters analysis
Through the S-parameter theory, by considering the reference impedance R 0 , one demonstrates that the Sparameters of the NGD topology presented in Figure 10 are expressed as: where ω ω is the impedance of the series RLC resonant network.The transmission phase of this microwave NGD topology is expressed as: One demonstrates that at the resonance frequency: the transmission parameter S 21 and the output return loss S 22 become respectively: Similarly to the NGD cell introduced in [14-16], the group delay will be also always negative at this resonance frequency:

Properties of the NGD circuit understudy
One can establish also that this microwave NGD value belongs in the frequency band corresponding to ,ω ω which is delimited by the following cut off frequencies: where It is interesting to note that the product of these NGD cut-off frequencies verify the following relation: But one points out that at very low frequencies 0 ≈ ω , the group delay is always positive.
This last expression leads us to the synthesis method of the NGD passive elements in the next paragraph.

Synthesis method
By inverting formulae (34) and (36) and the resonance frequency expression 0 ω , it yields the following synthesis relations: In this case, the output return loss S 22 will be systematically fixed to: Of course, for one cell, the gain S 21 can not be greater than T.Moreover, to provide a significant NGD with amplification in the wide bandwidth, the investigation of the cascaded structure is provided in the next subsection.
In order to verify the efficiency of the proposed theory, analysis of validation results will be proposed in the next section.

Numerical validation of the proposed NGD circuit for the modulated signals
The present design and simulation results were performed by using the microwave and electronic circuit simulator Advanced Design System (ADS) from Agilent TM .

Design of the microwave NGD device tested
The poof of concept to validate the former theoretic approach is comprised of the one-stage NGD active circuit shown in Figure 11.This later is designed with the LNA implemented in non-linear model using the package transistor AT-41411 provided by Agilent TM .Moreover, the considered passive components are from passive SMT models available in ADS library.The tested NGD device is biased at the DC voltage V b = 9 V and I b = 3.9 mA.

Frequency-domain analysis
Through the electromagnetic (EM) and circuit cosimulations in SPICE-Momentum ADS environment, the results obtained are sketched in Figure 12 and Figure 13.As illustrated in Figure 12, the tested circuit introduced in Figure 11 enables to generate a narrow-band NGD from 1.09 GHz to 1.26 GHz which can go down below -1.5 ns.Thus, one evaluates over the bandwidth of about 170 MHz.As displayed in Figure 13, in the considered frequency band, the insertion gain is above 0.5 dB and a good matching at the two accesses is achieved.

Analysis of the proposed NGD device non-linearity
To carry out a non-linear analysis of the circuit shown in Figure 11, harmonic balance simulations with the considerations of harmonics up to 11 th order were realized in the SPICE schematic environment of ADS.For that, a single tone harmonic signal with power P in_dBm at the single frequency f 0 = 1.18 GHz and varied from -15 dBm to 10 dBm was injected in the circuit tested.As consequence, the plot of the output power P out_dBm are obtained in function of the input level P i_ndBm depicted in Figure 14.So, one can see that the device understudy presents P in1dBC of about 5 dBm.Furthermore, by injecting two tones signal centred at the NGD centre frequency f 0 = 1.19 GHz with frequency shift of about 20 MHz.The input powers P in_dBm of these signals are set at around 5 dBm.After simulations, the result obtained is displayed in Figure 15.
Due to the non-linearity of the considered NGD circuit, for the first order harmonic with frequencies f 1 = 1.17 GHz and f 2 = 1.19 GHz, it appears P out (1 st order) ≈ -31 dBm and P out (3 rd order) ≈ -65 dBm in the operating frequency band.

Discussions
A microwave topology of active circuit susceptible to exhibit the NGD function with modulated signals is proposed in this section.The NGD topology investigated is comprised of an LNA in cascade with a shunt RLC-series network.The analysis of the proposed NGD topology Sparameters was conducted.The theoretic concept is then, completed by the establishment of synthesis methods of the NGD circuit parameters.This synthesis must be realized knowing the characteristics of the used amplifier and also the desired values of gain and NGD at the given specific frequency.The relevance of the theory was confirmed by the simulation results of one-stage NGD device designed with a hybrid LNA based on the package transistor AT-41411 from Agilent TM .Against the NGD active topology proposed in [15][16][17][18], through the present study, it is proved that a better access matching easily is achieved with a simpler biasing network.

General conclusions
A theory on the active topologies of electronic circuit susceptible to generate the NGD function is successfully investigated in this paper.Two different types of active topology were analyzed and synthesized.The first topology, mainly dedicated to the base band frequencies, is comprised of an RL-passive network in feedback with an RF/microwave amplifier.The other one, mainly operating with the modulated RF/microwave signals, is formed by an RLC-passive network associated in cascade with the amplifier.Theoretical analyses based on the S-parameters are presented by considering the gain and the matching access of the amplified used.Then, synthesis formulae for determining the elements R, L and C of the NGD circuits in function of the desired values of the gain and NGD are established.In addition, the basic properties of the NGD topologies investigated were also proposed.For checking the validity of the theoretical approaches, simulations both in frequency-and time-domains were performed.
With the base band NGD topology, NGD from DC up to tens MHz were realized accompanied with loss compensation.It was shown that the matching access levels of the proof of concept were widely lower than 0 dB up to 5 GHz.It was stated that the tested NFD circuit is unconditionally stable.Then, time-domain simulations confirm the functioning of the NGD circuit.It was observed that by injecting arbitrary wave form signals to the NGD circuit, it was demonstrated that the output signal wave fronts exit the circuit before those of the input penetrating in the circuit.As argued in [10][11][12][13][14][19][20], this effect is not in contradiction with the causality because the input bandwidth is sufficiently limited.Whereas with the second NGD topology investigated, NGD around 1.19 GHz with minimal value of about -1.5 ns was generated.It is shown that the proof of concept circuit presents good level of matching and excellent isolation loss.Finally, non-linear analyses were proposed to show the reaction of the NGD circuit with the power of the operating signals.
In the continuation of this work, the NGD topologies investigated will be integrated in the telecommunication systems for the enhancement of architectures notably for the delay cancellation.As ongoing research, design of circuit using MMIC LNA operating with base band signals up to RF/microwave frequencies is planned.The design of the NGD device capable to work up to several tens GHz based on distributed elements is also in progress.Lastly, the NGD concept is envisaged to be used for reducing the delays and also for enhancing the radio communication systems as reported in [45][46][47][48].

Figure 1 Figure 1 :
Figure 1 presents the topology of the NGD circuit understudy for the base band frequencies.It is comprised of an RL-network in feedback with an LNA A.

Figure 2 :
Figure 2: Prototype of two-stage NGD circuit for testing the time advance.

FrequencyFigure 3 :
Figure 3: Comparison of the group delay frequency responses from ideal RF circuit and realistic circuit with S2P models including the bias networks.

FrequencyFigure 4 :
Figure 4: Comparison of the group delay frequency responses from the ideal RF circuit and realistic circuit with S2P models including the bias networks.

Figure 5 :
Figure 5: Isolation and return losses of the NGD device shown in Figure 2.

Figure 10 presentsFigure 10 :
Figure10presents the microwave NGD circuit using LNA understudy in the present section.This unit cell of the understudy NGD circuit topology is composed of an amplifier in cascade with an RLC-series resonant network.

F
Figure 12: Group delay ) ( f τ of the designed NGD circuit tested.