Thông tin tài liệu
MicrowaveandMillimeterWaveTechnologies:ModernUWBantennasandequipment412 The proposed method has been also applied to predict the out-of-band response of the dual- mode filter just considered before. In Fig. 11, we compare the S parameters of such structure in a very wide frequency band (1000 points comprised between 8 and 18 GHz) when k 0 is chosen to be 0 and equal to the value of the in-band center frequency. As it can be observed, both results are slightly less accurate at very high frequencies (far from the center frequency of the filter), and more accuracy is preserved when additional terms are considered in (107), i.e. when k 0 0. In order to recover very accurate results in a very wide frequency band, it has been needed to increase the total number of poles to 75, thus involving a global CPU effort (1000 points) of 0.39 s. Fig. 11. Out-of-band response of the H-plane dual-mode filter. 7. Conclusions In this chapter, we have presented a very efficient procedure to compute the wideband generalized impedance and admittance matrix representations of cascaded planar waveguide junctions, which allows to model a wide variety of real passive components. The proposed method provides the generalized matrices of waveguide steps and uniform waveguide sections in the form of pole expansions. Then, such matrices are combined following an iterative algorithm, which finally provides a wideband matrix representation of the complete structure. Proceeding in this way, the most expensive computations are performed outside the frequency loop, thus widely reducing the computational effort required for the analysis of complex geometries with a high frequency resolution. The accuracy and numerical efficiency of this new technique have been successfully validated through the full-wave analysis of several waveguide filters. With regard to the ananlysis of single waveguide steps, the Z matrix representation offers a better computational efficiency than the Y matrix representation. However, our novel techniche for the admittance case can even provide a better performance than the original integral equation formulated in terms of the Z matrix. The Y matrix represention is useful when the devices under study include building blocks (i.e. arbitrarily shaped 3D cavities) whose analysis through the BI-RME method typically provides admittance matrices. In this way, a wideband cascade connection of Y matrices can be applied. 8. References Alessandri, F., G. Bartolucci, and R. Sorrentino (1988), Admittance matrix formulation of waveguide discontinuity problems: Computer-aided design of branch guide directional couplers, IEEE Trans. Microwave Theory Tech., 36(2), 394-403. Alessandri, F., M. Mongiardo, and R. Sorrentino (1992), Computer-aided design of beam forming networks for modern satellite antennas, IEEE Trans. Microwave Theory Tech., 40(6), 1117- 1127. Alvarez-Melcón, A., G. Connor, and M. Guglielmi (1996), New simple procedure for the computation of the multimode admittance or impedance matrix of planar waveguide junctions, IEEE Trans. Microwave Theory Tech., 44(3), 413-418. Arcioni, P., and G. Conciauro (1999), Combination of generalized admittance matrices in the form of pole expansions, IEEE Trans. Microwave Theory Tech., 47(10), 1990-1996. Arcioni, P., M. Bressan, G. Conciauro, and L. Perregrini (1996), Wideband modeling of arbitrarily shaped E-plane waveguide components by the boundary integral-resonant mode expansion method, IEEE Trans. Microwave Theory Tech., 44(11), 2083-2092. Arcioni, P., M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini (2002), Frequency/time- domain modeling of 3D waveguide structures by a BI-RME approach, Int. Journal of Numer. Modeling: Electron. Networks, Devices and Fields, 15(1), 3-21. Boria, V. E., and B. Gimeno (2007), Waveguide filters for satellites, IEEE Microwave Magazine, 8(5), 60-70. Boria, V. E., G. Gerini, and M. Guglielmi (1997), An efficient inversion technique for banded linear systems, in IEEE MTT-S Int. Microw. Symp. Digest, pp. 1567-1570, Denver. Conciauro, G., P. Arcioni, M. Bressan, and L. Perregrini (1996), Wideband modelling of arbitrarily shaped H-plane waveguide components by the boundary integral-resonant mode expansion method, IEEE Trans. Microwave Theory Tech., 44(7), 1057-1066. Conciauro, G., M. Guglielmi, and R. Sorrentino (2000), Advanced Modal Analysis - CAD Techniques for Waveguide Components and Filters, Wiley, Chichester. Eleftheriades, G., A. Omar, L. Katehi, and G. Rebeiz (1994), Some important properties of waveguide junction generalized scattering matrices in the context of the mode matching technique, IEEE Trans. Microwave Theory Tech., 42(10), 1896-1903. Gerini, G., M. Guglielmi, and G. Lastoria (1998), Efficient integral equation formulations for impedance or admittance representation of planar waveguide junction, in IEEE MTT-S Int. Microw. Symp. Digest, pp. 1747-1750, Baltimore. Guillot, P., P. Couffignal, H. Baudrand, and B. Theron, “Improvement in calculation of some surface integrals: Application to junction characterization in cavity filter design”, IEEE Trans. Microwave Theory Tech., vol. 41, no. 12, pp. 2156-2160, Dec. 1993. Gradstheyn, I. S., and I. M. Ryzhik (1980), Table of Integrals, Series and Products, Academic Press. Guglielmi, M., and A. Alvarez-Melcón (1993), Rigorous multimode equivalent network representation of capacitive discontinuities, IEEE Trans. Microwave Theory Tech., 41(6/7), 1195-1206. WidebandRepresentationofPassiveComponentsbasedonPlanarWaveguideJunctions 413 The proposed method has been also applied to predict the out-of-band response of the dual- mode filter just considered before. In Fig. 11, we compare the S parameters of such structure in a very wide frequency band (1000 points comprised between 8 and 18 GHz) when k 0 is chosen to be 0 and equal to the value of the in-band center frequency. As it can be observed, both results are slightly less accurate at very high frequencies (far from the center frequency of the filter), and more accuracy is preserved when additional terms are considered in (107), i.e. when k 0 0. In order to recover very accurate results in a very wide frequency band, it has been needed to increase the total number of poles to 75, thus involving a global CPU effort (1000 points) of 0.39 s. Fig. 11. Out-of-band response of the H-plane dual-mode filter. 7. Conclusions In this chapter, we have presented a very efficient procedure to compute the wideband generalized impedance and admittance matrix representations of cascaded planar waveguide junctions, which allows to model a wide variety of real passive components. The proposed method provides the generalized matrices of waveguide steps and uniform waveguide sections in the form of pole expansions. Then, such matrices are combined following an iterative algorithm, which finally provides a wideband matrix representation of the complete structure. Proceeding in this way, the most expensive computations are performed outside the frequency loop, thus widely reducing the computational effort required for the analysis of complex geometries with a high frequency resolution. The accuracy and numerical efficiency of this new technique have been successfully validated through the full-wave analysis of several waveguide filters. With regard to the ananlysis of single waveguide steps, the Z matrix representation offers a better computational efficiency than the Y matrix representation. However, our novel techniche for the admittance case can even provide a better performance than the original integral equation formulated in terms of the Z matrix. The Y matrix represention is useful when the devices under study include building blocks (i.e. arbitrarily shaped 3D cavities) whose analysis through the BI-RME method typically provides admittance matrices. In this way, a wideband cascade connection of Y matrices can be applied. 8. References Alessandri, F., G. Bartolucci, and R. Sorrentino (1988), Admittance matrix formulation of waveguide discontinuity problems: Computer-aided design of branch guide directional couplers, IEEE Trans. Microwave Theory Tech., 36(2), 394-403. Alessandri, F., M. Mongiardo, and R. Sorrentino (1992), Computer-aided design of beam forming networks for modern satellite antennas, IEEE Trans. Microwave Theory Tech., 40(6), 1117- 1127. Alvarez-Melcón, A., G. Connor, and M. Guglielmi (1996), New simple procedure for the computation of the multimode admittance or impedance matrix of planar waveguide junctions, IEEE Trans. Microwave Theory Tech., 44(3), 413-418. Arcioni, P., and G. Conciauro (1999), Combination of generalized admittance matrices in the form of pole expansions, IEEE Trans. Microwave Theory Tech., 47(10), 1990-1996. Arcioni, P., M. Bressan, G. Conciauro, and L. Perregrini (1996), Wideband modeling of arbitrarily shaped E-plane waveguide components by the boundary integral-resonant mode expansion method, IEEE Trans. Microwave Theory Tech., 44(11), 2083-2092. Arcioni, P., M. Bozzi, M. Bressan, G. Conciauro, and L. Perregrini (2002), Frequency/time- domain modeling of 3D waveguide structures by a BI-RME approach, Int. Journal of Numer. Modeling: Electron. Networks, Devices and Fields, 15(1), 3-21. Boria, V. E., and B. Gimeno (2007), Waveguide filters for satellites, IEEE Microwave Magazine, 8(5), 60-70. Boria, V. E., G. Gerini, and M. Guglielmi (1997), An efficient inversion technique for banded linear systems, in IEEE MTT-S Int. Microw. Symp. Digest, pp. 1567-1570, Denver. Conciauro, G., P. Arcioni, M. Bressan, and L. Perregrini (1996), Wideband modelling of arbitrarily shaped H-plane waveguide components by the boundary integral-resonant mode expansion method, IEEE Trans. Microwave Theory Tech., 44(7), 1057-1066. Conciauro, G., M. Guglielmi, and R. Sorrentino (2000), Advanced Modal Analysis - CAD Techniques for Waveguide Components and Filters, Wiley, Chichester. Eleftheriades, G., A. Omar, L. Katehi, and G. Rebeiz (1994), Some important properties of waveguide junction generalized scattering matrices in the context of the mode matching technique, IEEE Trans. Microwave Theory Tech., 42(10), 1896-1903. Gerini, G., M. Guglielmi, and G. Lastoria (1998), Efficient integral equation formulations for impedance or admittance representation of planar waveguide junction, in IEEE MTT-S Int. Microw. Symp. Digest, pp. 1747-1750, Baltimore. Guillot, P., P. Couffignal, H. Baudrand, and B. Theron, “Improvement in calculation of some surface integrals: Application to junction characterization in cavity filter design”, IEEE Trans. Microwave Theory Tech., vol. 41, no. 12, pp. 2156-2160, Dec. 1993. Gradstheyn, I. S., and I. M. Ryzhik (1980), Table of Integrals, Series and Products, Academic Press. Guglielmi, M., and A. Alvarez-Melcón (1993), Rigorous multimode equivalent network representation of capacitive discontinuities, IEEE Trans. Microwave Theory Tech., 41(6/7), 1195-1206. MicrowaveandMillimeterWaveTechnologies:ModernUWBantennasandequipment414 Guglielmi, M., and G. Gheri (1994), Rigorous multimode network representation of capacitive steps, IEEE Trans. Microwave Theory Tech., 42(4), 622-628. Guglielmi, M., and G. Gheri (1995), Multimode equivalent network representation of inductive and capacitive multiple posts, IEE Proc. Microwave Antennas Propag., 142(1), 41-46. Guglielmi, M., and C. Newport (1990), Rigorous multimode equivalent network representation of inductive discontinuities, IEEE Trans. Microwave Theory Tech., 38(11), 1651-1659. Guglielmi, M., G. Gheri, M. Calamia, and G. Pelosi (1994), Rigorous multimode network numerical representation of inductive step, IEEE Trans. Microwave Theory Tech., 42(2), 317-326. Gugliemi, M., P. Jarry, E. Kerherve, O. Roquebrum, and D. Schmitt (2001), A new family of all- inductive dual-mode filters, IEEE Trans. Microwave Theory Tech., 49(10), 1764-1769. Itoh, T. (1989), Numerical Techniques for Microwave and Millimeter-Wave Passive Structures, Wiley, New York. Lastoria, G., G. Gerini, M. Guglielmi, and F. Emma (1998), CAD of triple-mode cavities in rectangular waveguide, IEEE Microwave and Guided Wave Letters, 8(10), 339-341. Mansour, R., and R. MacPhie (1986), An improved transmission matrix formulation of cascaded discontinuities and its application to E-plane circuits, IEEE Trans. Microwave Theory Tech., 34(12), 1490-1498. Mira, F., A. A. San Blas, V. E. Boria, B. Gimeno, M. Bressan, and L. Perregrini (2006), Efficient pole-expansion of the generalized impedance matrix representation of planar waveguide junctions, in IEEE MTT-S Int. Microw. Symp. Digest, pp. 1033-1036, San Francisco. Safavi-Naini, R., and R. MacPhie (1981), On solving waveguide junction scattering problems by the conservation of complex power technique, IEEE Trans. Microwave Theory Tech., 29(4), 337-343. Safavi-Naini, R., and R. MacPhie (1982), Scattering at rectangular-to-rectangular waveguide junctions, IEEE Trans. Microwave Theory Tech., 30(11), 2060-2063. Sorrentino, R. (1989), Numerical Methods for Passive Microwave and Millimeter-Wave Structures, IEEE Press, New York. Spiegel, M. R. (1991), Complex Variables, Mc Graw-Hill. Uher, J., J. Bornemann, and U. Rosenberg (1993), Waveguide Components for Antenna Feed Systems: Theory and CAD, Artech-House, Norwood. Wexler, A. (1967), Solution of waveguide discontinuities by modal analysis, IEEE Trans. Microwave Theory Tech., 15(9), 508-517. Zhang, F., Matrix Theory, Springer-Verlag, New York, 1999. StudyandApplicationofMicrowaveActiveCircuitswithNegativeGroupDelay 415 StudyandApplicationofMicrowaveActiveCircuitswithNegativeGroup Delay BlaiseRavelo,AndréPérennecandMarcLeRoy x Study and Application of Microwave Active Circuits with Negative Group Delay Blaise Ravelo, André Pérennec and Marc Le Roy UEB France, University of Brest, Lab-STICC, UMR CNRS 3192, France 1. Introduction Since the early of 1970s, the interpretation of the negative group delay (NGD) phenomenon has attracted considerable attention by numerous scientists and physicists (Garrett & McGumber, 1970; Chu & Wong, 1982; Chiao et al., 1996; Mitchell & Chiao, 1997 and 1998; Wang et al., 2000). Several research papers devoted to the confirmation of its existence, in particular in electronic and microwave domains, have been published (Lucyszyn et al., 1993; Broomfield & Everard, 2000; Eleftheriades et al., 2003; Siddiqui et al., 2005; Munday & Henderson, 2004; Nakanishi et al., 2002; Kitano et al., 2003; Ravelo et al., 2007a and 2008a). In these papers, both theoretical and experimental verifications were performed. The NGD demonstrators that exhibit NGD or negative group velocity were based either on passive resonant circuits (Lucyszyn et al., 1993; Broomfield & Everard, 2000; Eleftheriades et al., 2003; Siddiqui et al., 2005) or on active ones (Chiao et al., 1996; Munday & Henderson, 2004; Nakanishi et al., 2002; Kitano et al., 2003). In practice, it was found that the investigated NGD passive circuits proved to be systematically accompanied with losses sometimes greater than 10 dB. While the active ones which use essentially classical operational amplifiers in feedback with R, L and C passive network were limited at only some MHz. Through experiments with these electronic active circuits, it was pointed out (Mitchell & Chiao, 1997 and 1998; Eleftheriades et al., 2003; Kitano et al., 2003) that the apparition of this counterintuitive phenomenon is not at odds with the causality principle. These limitations, i.e. losses and/or restriction on the frequency range drove us to develop the new NGD cell presented in Fig. 1 (Ravelo, Pérennec and Le Roy, 2007a, 2007b, 2007c, 2008a and 2008b). Fig. 1. NGD active cell and its low-frequency model; g m : transconductance and R ds : drain- source resistor. 21 MicrowaveandMillimeterWaveTechnologies:ModernUWBantennasandequipment416 This NGD active cell is composed of a field effect transistor (FET) terminated with a shunt RLC series network. We remark that this cell corresponds typically to the topology of a classical resistive amplifier, but here, focus is only on the generation of the NGD function together with gain. In this way, we recently demonstrated (Ravelo et al., 2007a, 2007b) that the group delay of this NGD cell is always negative at its resonance frequency, LC/1 0 . Furthermore, as FETs operating in different frequency ranges are available, the cell is potentially able to operate at microwave wavelengths and over broad bandwidths (Ravelo et al., 2007b). The a priori limitations rely on the operating frequency band of the lumped RLC components. By definition, the group delay is given by the opposite of the transmission phase, )()( 21 jS derivative with respect to the angular frequency, ω: )j(S )( 21 . (1) Analytical demonstrations and frequency measurements had previously allowed us to state that, in addition to NGD, this active cell can generate amplification with a good access matching. The present chapter is organized in two main sections. The fundamental theory permitting the synthesis of this NGD cell is developed in details in Section 2. Then, through a time domain study based on a Gaussian wave pulse response, the physical meaning of this phenomenon at microwave wavelengths is provided (Ravelo, 2008). From a second effect caused by the gain shape of the NGD active cell shown in Fig. 1, an easy method to realize pulse compression (PC) is also developed and examined. To illustrate the relevance and highlight the benefits of this innovative NGD topology, Section 3 deals with a new concept of frequency-independent active phase shifter (PS) used in recent applications (Ravelo et al., 2008b). This NGD PS is mainly composed of a positive group delay device set in cascade with a negative one of similar absolute value. To validate this innovative PS principle, a hybrid planar prototype was fabricated and tested. The measurements proved to well- correlate to the simulations and showed a phase flatness less than ±10° over a relative frequency band of 100%. Further to the use of two NGD cells, the results of the simulations run with a second PS showed an improvement of the relative constant-phase bandwidth up to about 125 %. These innovative PSs were also used to design and investigate a broadband active balun (Ravelo et al., 2007c). Finally, applications of this microwave NGD active device in telecommunication equipments are proposed, and further improvements are discussed. 2. Theoretical and experimental study of the proposed NGD active topology This Section deals with the analytical and experimental studies of the NGD active cell schematized in Fig. 1. After a brief recall of the S-parameters analysis, the synthesis relations appropriated to this cell are given in Subsection 2.1. Then, Subsection 2.2 is focused on the time-domain response of this cell in the case-study of a Gaussian input-wave pulse; the basic theory evidencing the associated pulse compression phenomenon is proposed. Subsection 2.3 is devoted to the description of experimental results obtained in both frequency- and time-domains; explanations about the process in use to design the NGD active device under test are also provided. 2.1 S-parameters analysis and synthesis relations As established in Ravelo et al., 2007a and 2007b, by using the low-frequency classical model of a FET, the scattering matrix of the ideal NGD cell presented in Fig. 1 is expressed as: 1)( 11 jS , (2) 0)( 12 jS , (3) )]RZ(ZRZ[ RgZZ )j(S dsds dsm 00 0 21 2 , (4) )RZ(ZRZ )RZ(ZZR )j(S dsds dsds 00 0 22 , (5) where )]/(1[ CLjRZ . (6) Z 0 is the port reference impedance, usually 50. At the resonance angular frequency, LC/1 0 , one gets Z = R, and then equations (4) and (5) become: )]RZ(RRZ[ RgRZ )(S dsds dsm 00 0 021 2 , (7) )RZ(RRZ )RR(ZRR )(S dsds dsds 00 0 022 . (8) At this frequency, it was demonstrated (Ravelo et al., 2007a and 2007b) that the group delay expressed in equation (1) is always negative: )]([ 2 )( 0 0 0 dsds ds RRZRRR RLZ . (9) The synthesis relations relative to the NGD cell are extracted from equations (8) and (9). As shown hereafter, they depend on the given gain magnitude and group delay (S 21 and τ 0 , respectively) at the resonance, ω 0 : ])RZ(SRZg[ RZS R dsdsm ds0 0210 21 2 , (10) )RZ( )]RR(ZRR[R L ds dsds 0 00 2 . (11) Then, the C synthesis relation is deduced from the expression of the resonance angular frequency: )L( C 2 0 1 . (12) StudyandApplicationofMicrowaveActiveCircuitswithNegativeGroupDelay 417 This NGD active cell is composed of a field effect transistor (FET) terminated with a shunt RLC series network. We remark that this cell corresponds typically to the topology of a classical resistive amplifier, but here, focus is only on the generation of the NGD function together with gain. In this way, we recently demonstrated (Ravelo et al., 2007a, 2007b) that the group delay of this NGD cell is always negative at its resonance frequency, LC/1 0 . Furthermore, as FETs operating in different frequency ranges are available, the cell is potentially able to operate at microwave wavelengths and over broad bandwidths (Ravelo et al., 2007b). The a priori limitations rely on the operating frequency band of the lumped RLC components. By definition, the group delay is given by the opposite of the transmission phase, )()( 21 jS derivative with respect to the angular frequency, ω: )j(S )( 21 . (1) Analytical demonstrations and frequency measurements had previously allowed us to state that, in addition to NGD, this active cell can generate amplification with a good access matching. The present chapter is organized in two main sections. The fundamental theory permitting the synthesis of this NGD cell is developed in details in Section 2. Then, through a time domain study based on a Gaussian wave pulse response, the physical meaning of this phenomenon at microwave wavelengths is provided (Ravelo, 2008). From a second effect caused by the gain shape of the NGD active cell shown in Fig. 1, an easy method to realize pulse compression (PC) is also developed and examined. To illustrate the relevance and highlight the benefits of this innovative NGD topology, Section 3 deals with a new concept of frequency-independent active phase shifter (PS) used in recent applications (Ravelo et al., 2008b). This NGD PS is mainly composed of a positive group delay device set in cascade with a negative one of similar absolute value. To validate this innovative PS principle, a hybrid planar prototype was fabricated and tested. The measurements proved to well- correlate to the simulations and showed a phase flatness less than ±10° over a relative frequency band of 100%. Further to the use of two NGD cells, the results of the simulations run with a second PS showed an improvement of the relative constant-phase bandwidth up to about 125 %. These innovative PSs were also used to design and investigate a broadband active balun (Ravelo et al., 2007c). Finally, applications of this microwave NGD active device in telecommunication equipments are proposed, and further improvements are discussed. 2. Theoretical and experimental study of the proposed NGD active topology This Section deals with the analytical and experimental studies of the NGD active cell schematized in Fig. 1. After a brief recall of the S-parameters analysis, the synthesis relations appropriated to this cell are given in Subsection 2.1. Then, Subsection 2.2 is focused on the time-domain response of this cell in the case-study of a Gaussian input-wave pulse; the basic theory evidencing the associated pulse compression phenomenon is proposed. Subsection 2.3 is devoted to the description of experimental results obtained in both frequency- and time-domains; explanations about the process in use to design the NGD active device under test are also provided. 2.1 S-parameters analysis and synthesis relations As established in Ravelo et al., 2007a and 2007b, by using the low-frequency classical model of a FET, the scattering matrix of the ideal NGD cell presented in Fig. 1 is expressed as: 1)( 11 jS , (2) 0)( 12 jS , (3) )]RZ(ZRZ[ RgZZ )j(S dsds dsm 00 0 21 2 , (4) )RZ(ZRZ )RZ(ZZR )j(S dsds dsds 00 0 22 , (5) where )]/(1[ CLjRZ . (6) Z 0 is the port reference impedance, usually 50. At the resonance angular frequency, LC/1 0 , one gets Z = R, and then equations (4) and (5) become: )]RZ(RRZ[ RgRZ )(S dsds dsm 00 0 021 2 , (7) )RZ(RRZ )RR(ZRR )(S dsds dsds 00 0 022 . (8) At this frequency, it was demonstrated (Ravelo et al., 2007a and 2007b) that the group delay expressed in equation (1) is always negative: )]([ 2 )( 0 0 0 dsds ds RRZRRR RLZ . (9) The synthesis relations relative to the NGD cell are extracted from equations (8) and (9). As shown hereafter, they depend on the given gain magnitude and group delay (S 21 and τ 0 , respectively) at the resonance, ω 0 : ])RZ(SRZg[ RZS R dsdsm ds0 0210 21 2 , (10) )RZ( )]RR(ZRR[R L ds dsds 0 00 2 . (11) Then, the C synthesis relation is deduced from the expression of the resonance angular frequency: )L( C 2 0 1 . (12) MicrowaveandMillimeterWaveTechnologies:ModernUWBantennasandequipment418 As previously mentioned, in addition to this NGD property, this circuit allows compression of the width of a modulated Gaussian pulse centred at ω 0 . The compression theory will be developed in the next section. 2.2 Study of the Gaussian-pulse response: evidence of time domain advance and pulse compression (PC) Fig. 2 illustrates the configuration under consideration in the time-domain study. It consists of a black box of the NGD circuit S-parameters excited by a sine carrier, f 0 = ω 0 /(2π), modulated by a Gaussian pulse. In order to evidence the principle of this NGD phenomenon, let us consider the input signal expressed as: tj T tt eetx x 0 2 2 0 2 )( )( . (13) Fig. 2. Block diagram of the understudied configuration considered in time-domain (Z 0 = 50 Ω). ∆T x is the standard deviation (half width at 1/e of the maximal input value) and t 0 is the central time of the Gaussian pulse. It ensues that the Fourier transform of such a signal is defined as: 00 2 0 2 )()(5.0 2)( tjT x x eTjX . (14) According to the signal processing theory, this function is also Gaussian, and its angular frequency standard deviation is: xx T /1 . (15) It means that the pulse compression in time domain involves a pulse expansion in frequency domain and vice versa. Then, the standard deviation of the Gaussian output is compared, at first theoretically, with the input one through the transmittance, H(jω). To highlight this analytical approach, let us consider the black box system shown in Fig. 2. As its transfer function, H(jω), is excited by X(jω), the output Fourier transform is: )()()()( )()(ln jXejXjHjY jHjArgjH . (16) A simplified and approximated analytical study is proposed hereafter in order to analyze the behaviour of this output. First, we consider the Taylor series expansion of the magnitude, ln|H(jω)|, around the resonant angular frequency, ω 0 : ])[()( )(2 )( )( )( )( )(ln)(ln 3 0 2 0 0 0 0 0 0 0 O H H H H HjH a , (17) where H’(ω 0 ) and H“(ω 0 ) are respectively the first- and second-order derivatives of |H(jω)| with respect to ω. Then, application of the same procedure to the transmission phase, )()( 21 jS leads to the following approximated expression: ])[()(5.0)()( ])[())((5.0))(()()( 3 0 2 00000 3 0 2 00000 O O a . (18) At resonance, φ(ω 0 ) = 0, and in the vicinity of ω 0 , one gets τ’(ω 0 ) ≈ 0, this implies that: ])[()()( 3 000 O a . (19) One should note that the terms of higher order can be ignored if the input signal bandwidth is small enough compared to the NGD bandwidth. As this phase response is relatively linear, the Y(jω)-magnitude is unaffected. So, the output modulus can be written as: )()()( 2 0 0 0 0 0 0 )( )(2 )( )( )( )( 0 jXeHjY H H H H . (20) By substituting for X(jω) from equation (14) in equation (20), one gets: 2 0 0 0 2 0 0 0 )( )( )( 2 1 )( )( )( 0 2)()( H H T H H x x eTHjY . (21) The magnitude of the insertion gain is defined as: 22 0 2 00 22 0 )]/(1[()()]([ )]/(1[(2 )( CLRZRZRRZ CLRRgZ jH dsdsds dsm . (22) At ω 0 , this expression becomes: )RR(ZRR RZRg )(H dsds dsm 0 0 0 2 , (23) and the first- and second-order derivatives are expressed as follows: 0 )( )( 0 0 jH H , (24) StudyandApplicationofMicrowaveActiveCircuitswithNegativeGroupDelay 419 As previously mentioned, in addition to this NGD property, this circuit allows compression of the width of a modulated Gaussian pulse centred at ω 0 . The compression theory will be developed in the next section. 2.2 Study of the Gaussian-pulse response: evidence of time domain advance and pulse compression (PC) Fig. 2 illustrates the configuration under consideration in the time-domain study. It consists of a black box of the NGD circuit S-parameters excited by a sine carrier, f 0 = ω 0 /(2π), modulated by a Gaussian pulse. In order to evidence the principle of this NGD phenomenon, let us consider the input signal expressed as: tj T tt eetx x 0 2 2 0 2 )( )( . (13) Fig. 2. Block diagram of the understudied configuration considered in time-domain (Z 0 = 50 Ω). ∆T x is the standard deviation (half width at 1/e of the maximal input value) and t 0 is the central time of the Gaussian pulse. It ensues that the Fourier transform of such a signal is defined as: 00 2 0 2 )()(5.0 2)( tjT x x eTjX . (14) According to the signal processing theory, this function is also Gaussian, and its angular frequency standard deviation is: xx T /1 . (15) It means that the pulse compression in time domain involves a pulse expansion in frequency domain and vice versa. Then, the standard deviation of the Gaussian output is compared, at first theoretically, with the input one through the transmittance, H(jω). To highlight this analytical approach, let us consider the black box system shown in Fig. 2. As its transfer function, H(jω), is excited by X(jω), the output Fourier transform is: )()()()( )()(ln jXejXjHjY jHjArgjH . (16) A simplified and approximated analytical study is proposed hereafter in order to analyze the behaviour of this output. First, we consider the Taylor series expansion of the magnitude, ln|H(jω)|, around the resonant angular frequency, ω 0 : ])[()( )(2 )( )( )( )( )(ln)(ln 3 0 2 0 0 0 0 0 0 0 O H H H H HjH a , (17) where H’(ω 0 ) and H“(ω 0 ) are respectively the first- and second-order derivatives of |H(jω)| with respect to ω. Then, application of the same procedure to the transmission phase, )()( 21 jS leads to the following approximated expression: ])[()(5.0)()( ])[())((5.0))(()()( 3 0 2 00000 3 0 2 00000 O O a . (18) At resonance, φ(ω 0 ) = 0, and in the vicinity of ω 0 , one gets τ’(ω 0 ) ≈ 0, this implies that: ])[()()( 3 000 O a . (19) One should note that the terms of higher order can be ignored if the input signal bandwidth is small enough compared to the NGD bandwidth. As this phase response is relatively linear, the Y(jω)-magnitude is unaffected. So, the output modulus can be written as: )()()( 2 0 0 0 0 0 0 )( )(2 )( )( )( )( 0 jXeHjY H H H H . (20) By substituting for X(jω) from equation (14) in equation (20), one gets: 2 0 0 0 2 0 0 0 )( )( )( 2 1 )( )( )( 0 2)()( H H T H H x x eTHjY . (21) The magnitude of the insertion gain is defined as: 22 0 2 00 22 0 )]/(1[()()]([ )]/(1[(2 )( CLRZRZRRZ CLRRgZ jH dsdsds dsm . (22) At ω 0 , this expression becomes: )RR(ZRR RZRg )(H dsds dsm 0 0 0 2 , (23) and the first- and second-order derivatives are expressed as follows: 0 )( )( 0 0 jH H , (24) MicrowaveandMillimeterWaveTechnologies:ModernUWBantennasandequipment420 0 )( )(28 )( )( 3 0 00 22 0 2 2 2 0 0 dsds dsdsdsm RRZRRR RZRZRLZRg jH H . (25) It ensues that the output amplitude of equation (21) can be simplified as follows: 2 000 2 ))]((/)([5.0 0 2)()( HHT x x eTHjY . (26) In fact, thanks to the second-order expansion expressed in equation (26), the output Fourier transform also behaves as a Gaussian pulse: )2/()( max 22 0 )( y eYjY , (27) of amplitude: 2 )( 2 0 0 max x dsds dsm T RRZRR RZRg Y . (28) and with an angular frequency standard deviation such as: ])(/[)(1)(/)( 1 2 0000 2 x x x y THHHHT . (29) Furthermore, the pulse width is expanded in the frequency domain (∆ω x > ∆ω y ). In the time domain, the approximated output signal inferred from the inverse Fourier transform of equation (26) is written as: tj HHT tt x x ee HHT TH ty x 0 00 2 2 00 )(/)( )]([5.0 00 2 0 )(/)( )( )( . (31) It can be seen that this output behaves as a modulated Gaussian that exhibits a time advance whenever τ(ω 0 ) < 0; moreover, the standard deviation is expressed as: 2 0 2 000 2 2 )( )(24 dsds dsdsds xy RRZRRR RZRZRZRL TT . (32) This is obvisously an approximated expression because it comes from a first-order limited expansion of equation (17). Due to the intrinsic behaviour of linear devices, the higher order terms ensure that ∆T y cannot tend to zero. Hence, in practice, equation (32) corresponds to a compression of the pulse width in the time domain. Furthermore, compared to the input pulse, x(t), the output one, y(t) is amplified by the quantity: yx TTH /.)( 0 . (33) As reported by Cao and co-workers (2003), if ∆T x is getting closer to ∆T xmin , it goes along with a significant PC. Remarks on the PC phenomenon: Various PC techniques have been developed at optical- and microwave-wavelengths in order to convert a long-duration pulse into a shorter one. One should note that the principles and methods proposed in the literature depend on whether the applications under study are dedicated to low or high power (Gaponov- Grekhov & Granatstein, 1994; Thumm & Kasparek, 2002). For example, PC was investigated in ultra-fast laser systems (Li et al., 2005), then its use has become more and more common thanks to the development of chirped pulse amplification (Arbore, 1997; Wang & Yao; 2008a and 2008b). The next step was the compression of a pulse in a Mach- Zehnder-interferometer geometry achieved by passing a broadband ultra-short pulse through two chirped fibre Bragg gratings with different chirp rates (Zeitouny et al., 2005). In radar and communication systems, PC has been used to enhance the range resolution. In order to elevate microwave power, investigations by several authors have been focused on a microwave pulse compressor based on a passive resonant cavity (Burt et al., 2005; Baum, 2006). The prerequisites are that the compressor cavity must present a high Q-factor; in addition, the constituting waveguides should operate with an oversized mode of the field in order to increase the power strength, and the power microwave sources should be narrow-band. This set of requirements is met by quasi-optical cavities, particularly by the ring-shaped multi-mirror ones, where the energy is sent to the cavity via corrugated mirror (Kuzikov et al., 2004). But, as in practice the implementation of such a technique is usually very complex and at high cost, a new and much simpler PC technique based on the use of NGD structure was recently proposed (Cao et al., 2003) by using a classical operational amplifier. Though application of this technique to rather low power devices remains possible, one should be aware that it is intrinsically restricted to low frequencies. The PC method proposed here is close to this latter study, but it is able to operate in the microwave frequency range. All of these predictions from theory have to be experimentally verified; but prior to the discussion of the corresponding experimental results, it is worth describing the design process of the NGD devices under study. 2.3 Experimental study (a) Design process: The flow chart displayed in Fig. 3 lists the sequence of actions to be followed to design NGD active circuits. One should note that the proposed process is well-suited to the use of classical circuit simulator/designer software such as, for example, ADS software from Agilent TM . [...]... measurements: Figs 5(a) and 5(b) describe the results of the frequency measurements made with an EB364A Agilent Vector Network Analyzer 424 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment Fig 5 Measured results: (a) insertion gain/group delay, and (b) return losses Fig 6 Wide band frequency responses of the tested NGD device: (a) return losses and (b) S21-parameter and isolation... (a) and the phase (b) when R = 68 and C = 5 pF Effects of C variations: Contrarily to the two previous cases, Figs 13 show that the form of the frequency responses (gain and phase) are rather unaffected by C-variation On the other hand, it has a significant impact on both the centre frequency and the constant phase value 430 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment. .. approximated to a Gaussian pulse with a frequency standard deviation of about ∆fy ≈ 50 MHz (or ∆Tx ≈ 3.3 ns) In frequency domain, the output pulse 426 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment width is thus expanded of about 125% and amplified of about 3.86 dB with respect to the input pulse one Fig 8(a) Simulated spectra of the input and output voltage (Fourier transforms)... with ADS and by using S-probe components, the stability of this PS was analyzed by checking that the magnitude of the input and output reflection coefficients of the FET was kept below one from 1 GHz up to 10 GHz Fig 15( a) Comparisons of the transmission phase/group delay values obtained by simulations and measurements 432 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment. .. operating bandwidth of constant phase, 434 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment an easy way to achieve loss compensation or even an amplification thanks to the active circuit characteristics, the FET non-reciprocity entails a good isolation but this may be a drawback for certain applications, and the proposed topology provides a constant transmission phase and. .. ports In addition, the return losses |S11|dB and |S22|dB are better than 11 dB from 3.0 to 6.0 GHz, and the output return loss |S33|dB is kept over 436 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment 9 dB Finally, the differential flatness, ||S31|dB -|S21|dB|, is less than 1.1 dB from 3.5 to 6.0 GHz (a) (b) Fig 20 Phases (a) and magnitudes (b) of the balun simulated S-parameters... AgilentTM 422 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment Fig 3 Flow chart of the NGD-device design process (n is the number of NGD cells) The technique used to design these devices is similar to the one developed in the case of classical microwave devices (filter, amplifier, coupler …) During the synthesis of the circuit under study, focus is on the NGD level and the gain... Int Symp on Phased Array Syst Technol., pp 68-73, Oct 15- 18, 1996, Boston, MA Zeitouny, A.; Stepanov, S.; Levinson, O & Horowitz, M (2005) Optical generation of linearly chirped microwave pulses using fiber Bragg gratings, IEEE Photon Technol Lett., Vol 17, No 3, pp 660-662 440 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment ... (2005) A broadband Wilkinson balun using microstrip metamaterial lines, IEEE Ant Wireless Propagation Lett., Vol 4, Issue 1, pp 209-212 Arbore, M A (1997) Engineerable compression of ultrashort pulses by use of secondharmonic generation in chirped-period-poled lithium niobate, Opt Lett., Vol 22, No 17, pp 1341-1343 438 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment Baum,... previously argued, in the ideal case of a perfect impedance matching, the total phase response, Φt is merely the sum of Φp and Φn This implies the total phase response: 428 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment t ( f ) 2 ( n p )( f f 1 ) [ p ( f 1 ) n ( f 1 )] (36) Since the operating principle implies that the group delay absolute values are equal . discontinuities, IEEE Trans. Microwave Theory Tech., 41(6/7), 1195-1206. Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment 414 Guglielmi, M., and G. Gheri (1994), Rigorous. NGD active cell and its low-frequency model; g m : transconductance and R ds : drain- source resistor. 21 Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment 416 This. and the first- and second-order derivatives are expressed as follows: 0 )( )( 0 0 jH H , (24) Microwave and Millimeter Wave Technologies: Modern UWB antennas and equipment 420
Ngày đăng: 21/06/2014, 11:20
Xem thêm: Micowave and Millimeter Wave Technologies Modern UWB antennas and equipment Part 15 pdf
