Electrical and Computer Engineering Volume, Article ID 33864, 5 pages doi:.55//33864 Research Article Extended Composite Right/Left-Handed Transmission Line and Dual-Band Reactance Transformation Yuming Zhang and Barry Spielman Kunshan Industrial Technology Research Institute, 4F Technology Building, 666 South Weicheng Road, Kunshan, Jiangsu 5347, China Department of Electrical and Systems Engineering, Washington University in St. Louis, One Brookings Drive, Box 7, St. Louis, MO 633, USA Correspondence should be addressed to Yuming Zhang, yumingzhang@live.com Received 6 July 9; Accepted 3 October 9 Academic Editor: Hakan Kuntman Copyright Y. Zhang and B. Spielman. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited. An extended composite right/left-handed transmission line is introduced, and its dual-band bandpass filter characteristics are explored. Novel reactance transformations, derived from this transmission line, are formulated to transform a lo-pass prototype filter into a dual-band bandpass filter ith arbitrary dual pass bands, ell-defined in-band attenuation ripples, and high out-of-band rejection. The physical insight into such a dual-band bandpass filter is provided ith a dispersion analysis. The transformations are verified by simulated results for dual-band bandpass filters.. Introduction An extended composite right/left-handed (CRLH) transmission line is first introduced. Then, its fundamental properties are investigated. Consequently, a reactance transformation is derived for dual-band filter synthesis. Although a dual-band bandpass filter design using the reactance transformation stems from the extended CRLH transmission line concept, it is totally different from the approach used by Tseng and Itoh []. In their approach, a traditional bandpass filter is first designed using quarterave short circuited; then the dual-band bandpass filter ith arbitrary to pass bands is implemented by replacing the microstrip lines ith the CRLH transmission lines; the separation of these to pass bands is determined by the nonlinear phase slope of the CRLH tansmission lines. In our approach, a filter ith arbitrary dual pass bands is directly transformed from a lo-pass filter prototype; then the dual-band bandpass filter can be realized using traditional or metamaterial transmission lines; the separation of the to pass bands is precisely controlled by the reactance transformation. In addition, Guan et al. reported dual-band bandpass filters employing to successive frequency transformations []. In their approach, the dual-band bandpass filter is also similar to the extended CRLH transmission line. The center frequencies of to pass bands are used to define frequency transformations; that is, the to pass bands are positioned by the center frequencies and their boundaries cannot be accurately specified. Additionally, their frequency transformations can only be used in a narro band filter. Hoever, our reactance transformation is parameterized by edge frequencies, and the bands are precisely delimited by them. Also, our reactance transformation can be used in an arbitrary band filter.. Extended CRLH Transmission Line Figure shos the equivalent circuit model for an extended CRLH transmission line. It consists of the series resonator comprised of L and C (Resonator ), the series shunt resonator comprised of L 3 and C 3 (Resonator 3), the shunt shunt resonator comprised of L and C (Resonator ), and the shunt series resonator comprised of L 4 and C 4 (Resonator
Electrical and Computer Engineering Resonator L C Resonator 3 L 3 C 3 Resonator 4 d L 4 C 4 L Resonator Figure : A lumped-element model unit cell of an extended CRLH transmission line. {d is the length of the unit cell. L, L, L 3,and L 4 are the inductances in henries/m. C, C, C 3,andC 4 are the capacitances in farads/m.} 4). The elimination of Resonators 3 and 4 reduces the extended CRLH transmission line into a CRLH transmission line [3]. The extended CRLH transmission line becomes a dual CRLH transmission line [4] ithout Resonators and. This paper focuses only on an extended CRLH transmission line that satisfies to conditions. The first condition is that the four resonators in Figure have the same resonant frequency ω = L C = L 3 C 3 = L C = L 4 C 4, () here ω is the center frequency. The series impedance Z and shunt admittance Y in Figure are Z = jωl + jωc + Y = jωc + jωl + C L C 3 L 4 C jωc 3 +/jωl 3, () jωl 4 +/jωc 4. (3) The second condition requires that the resonant frequencies of Z and Y coincide at ω : (ω = L C 3 = L 4 C. (4) If either of the to conditions is not satisfied, three or more pass bands can occur... Auxiliary CRLH Transmission Line. Here, an auxiliary CRLH transmission line is introduced to simplify the analysis of the extended CRLH transmission line. The auxiliary CRLH transmission line is comprised of the elements L, C 3, L 4,andC, as shon in the inset of Figure. From(4), it is knon that the series and shunt resonant frequencies coincide at ω. The high-pass cutoff frequency ω UPPER and lo-pass cutoff frequency ω LOWER are expressed as follos [5]: ( ω UPPER ) = { ( ω + ( ) [ ( + ω + ( } ) ] ( ) 4 ω 4 = / ( ω +/ ( ω L) [/ ( ) ω +/ ( ω L ) ], ( ) 4/ ω 4 ( ω LOWER ) = = { ( ω + ( ) [ ( + ω + ( } ) ] ( ) 4 ω 4 / ( ω +/ ( ω L) [/ ( + ω +/ ( ω L ) ], ( ) 4/ ω 4 (5) here () = 4/L C and (ω L) = /4L 4 C 3.From(5), three frequency relationships are obtained as follos: = ω UPPER ω LOWER, (6) ω L = ω LOWER ω UPPER, (7) ω UPPERω LOWER = ω Lω R = ( ω. (8).. Dispersion of the Extended CRLH Transmission Line. The dispersion analysis for the extended CRLH transmission line is presented here. Using the ABCD matrix of the toport netork described in Figure and the Bloch-Floquet theorem, the propagation constant γ is determined by γ = α = d arccos ( ZY + ), if 4 <ZY<, (9) γ = jβ = j ( ) ZY + d arccosh, if ZY < 4 or ZY >, ( here Z and Y aredefinedin() and(3), respectively. Equation (9) corresponds to an evanescent ave hereas ( represents a propagating ave. ZY can be ritten as the follos: ( ) ZY = 4 ω L, () here ( ω = ω ω ). () ω ω and ω L are defined in (6) and(7), respectively. ZY = gives to resonant frequencies: esupper = [ 4ω + ( ] ω + ω, esloer = [ 4ω + ( ] (3) ω ) ω.
Electrical and Computer Engineering 3 ZY = 4 yields four cutoff frequencies: ω UU = [ 4ω 4 + ( ] ω UPPER) + ω UPPER, ω UL = [ 4ω 4 + ( ] ω LOWER) + ω LOWER, ω LU = [ 4ω 4 + ( ] ω LOWER ) ω LOWER, ω LL = [ 4ω 4 + ( ] ω UPPER ) ω UPPER. These frequencies are related as follos: (4) ω = esupperesloer = ω UU ω LL = ω UL ω LU. (5) Equations (5) and(8) can be used to sho that ω UPPER > ω >ω LOWER. Thus, it follos from (3)and(4) that ω UU >esupper >ω UL >ω >ω LU >esloer >ω LL. (6) Based upon (9) and(, the dispersion of the extended CRLH transmission line is plotted in Figure (a). It is noted that the four cutoff frequencies define key features of the dual passband behavior. ω LL and ω LU are the loer and upper edges of the loer pass band, respectively. ω UL and ω UU are the loer and upper edges of the upper pass band, respectively. A ave exhibits phase-backard propagation hen the operating frequency is beteen ω LL and esloer, or beteen ω UL and esupper. Hoever, a ave exhibits phase-forard propagation hen the operating frequency is beteen esloer and ω LU,orbeteenesUpper and ω UU. Otherise, the ave is evanescent. It is noteorthy that esloer and esupper are transition points beteen forard and backard aves. The performance beteen esloer and esupper is similar to that of a dual CRLH transmission line [4] hereas the performance belo esloer or greater than esupper is similar to that of a CRLH transmission line [3, 5]. Figure (b) represents the equivalent circuit behavior of the extended CRLH transmission line at different frequencies. When the operating frequency is less than esloer, the extended CRLH transmission line is equivalent to a pure left-handed transmission line. Thus, the high-pass cutoff frequency ω LL appears. When the operating frequency is beteen esloer and ω, the extended CRLH transmission line is equivalent to a pure right-handed transmission line ith the lo-pass cutoff frequency ω LU. When the operating frequency is beteen ω and esupper, the extended CRLH transmission line is equivalent to a pure left-handed transmission line ith the high-pass cutoff frequency ω UL. When the operating frequency is greater than esupper, the extended CRLH transmission line is equivalent to a pure right-handed transmission line ith the lo-pass cutoff frequency ω UU..3. Important Frequency Expressions. Using (6), (7), and (4), ω R and ω L are reritten by ω R = (ω UU ω LL ) (ω UL ω LU ), (7) ω L = (ω UL ω LU ) (ω UU ω LL ). (8) Equations (7), (8),and (5) are important expressions that are used in Section 3 to transform frequency and elements. It should be noted that all formulations presented in Section are derived from the infinite extended CRLH transmission line. In practice, no transmission line is infinite. Hoever, it has been verified in [3] that the infinitestructure approximation provides reasonable accuracy hen asufficiently large number (>3) cells are used. 3. Dual-Band Reactance Transformation In this section, the dual-band frequency and element transformations are derived. The frequency transformation is a reactance function and converts a lo-pass prototype filter ith ell-defined insertion loss [6] to a dual-band bandpass filter. Simulation results are presented to verify the transformations. 3.. Dual-Band Bandpass Frequency Transformations. The desired frequency transformation is obtained by replacing the reactance and the susceptance in the prototype filter by series and shunt resonators such that jωl = Z, (9) jωc = Y, ( here Ω, L and C are the normalized frequency, inductance, and capacitance of the lo-pass prototype filter. Z and Y are defined in () and(3), respectively. Multiplying the right and left sides of (9) and(, respectively, the dual-band bandpass frequency transformation is obtained as follos: ( Ω = ) ω L, () here the definitions of, and ω L are given in (), (7) and(8), respectively. The normalized lo-pass cutoff frequency, Ω R = 4/LC =, of the pure right-handed transmission line is used for the derivation of (). With the substitution of, the explicit expression of / shos it is a lo-pass to bandpass transformation ith the passband bandidth of [6]. ω L/ is a lo-pass to bandstop transformation ith the stopband bandidth of ω L [6]. The frequency mappings of () are depicted in Figure 3. Itis observed that the subtraction of the lo-pass to bandstop mapping from the lo-pass to bandpass mapping gives the lo-pass to dual-band bandpass mapping.
4 Electrical and Computer Engineering.5.5 ω UU esupper esloer ω UL ω ωlu ω LL.5 π π βd, αd (radian) ω ω UU esupper ω UL ω ω LU esloer ω LL αd βd (a) (b) Figure : The dispersion (a) and equivalent circuits (b) of the extended CRLH transmission line (β and αare defined in (9) and (.) Normalised radian frequency Ω 5 4 3 3 4 5 /ω R ω L / /ω R ω L / ω LL ω LU ω UL ω UU Radian frequency ω Attenuation (db) Attenuation (db) 6 5 4 3.5 5 4 3.6.8..4.6.8..4 3 5 4.9.95.5. Attenuation (db).5 5 4 3 3 5 4.8.9. Figure 4: Simulated attenuation responses of the dual-band bandpass filters transformed from -db Chebyshev lo-pass filters for filter orders of 3, 4, and 5. (The response of the loer and upper pass bands in the upper subplot is amplified in the left and right bottom subplots, resp.) Figure 3: Mapping from normalized frequency of a prototype lopass filter into actual filter frequency. (The solid line, plotted by Ω = /ω R ω L/, is for the dual-band bandpass filter. The circle and dot lines describe the bandstop transformation ith minus sign, Ω = ω L/, and the bandpass transformation, Ω = /ω R,resp.Here, = ω (ω/ω ω /ω).) It is observed that L, C, L,andC constitute a bandpass filter ith the center frequency ω and passband bandidth of ω R. L 3, C 3, L 4,andC 4 form a band stop filter ith center frequency ω and a stop band bandidth of ω L. 3.. The Element Transformations. Substituting Ω ith () in (9) and(, and using (), the element transformations areobtainedasfollos: L = L, C = L ω, C = C, L = C ω, C 3 =, L Lω L 3 = C 3 ω, L 4 =, C Cω L 4 = L 4 ω. () 3.3. Simulations. Dual-band bandpass filter, specified by the band edge frequencies of.9 GHz,. GHz,.78 GHz, and. GHz, as transformed from a Chebyshev lo-pass filter ith -db passband ripple using (). Figure 4 shos simulated attenuation responses of lumped-element dualband bandpass filters of orders 3, 4, and 5. The loer and upper pass bands have the same ripple magnitude. The attenuation tends to infinity at the center frequency ω =.375 GHz. The observed pass band center frequencies,.ghz and.87 GHz, agree ith the computed esloer and esupper,respectively.
Electrical and Computer Engineering 5 4. Conclusion The extended CRLH transmission line has been introduced and characterized. Novel dual-band bandpass reactance transformations ere derived. A dual structure of the extended CRLH transmission line can be explored using an approach similar to that described here. A dual-band bandstop frequency transformation can be obtained by replacing Ω by /Ω. References [] C.-H. Tseng and T. Itoh, Dual-band bandpass and bandstop filters using composite right/left-handed metamaterial transmission lines, in Proceedings of the IEEE MTT-S International Microave Symposium Digest, pp. 93 934, San Francisco, Calif, USA, June 6. [] X. H. Guan, Z. Ma, P. Cai, et al., A dual-band bandpass filter synthesized by using frequency transformation and circuit conversion technique, in Proceedings of the IEEE Asia-Pacific Microave Conference (APMC 5), vol. 4, Yokohama, Japan, December 5. [3] C. Caloz and T. Itoh, Electromagnetic Metamaterial; Transmission Line Theory and Microave Applications: The Engineering Approach, John Wiley & Sons, Hoboken, NJ, USA, 6. [4] C. Caloz, Dual composite right/left-handed (D-CRLH) transmission line metamaterial, IEEE Microave and Wireless Components Letters, vol. 6, no., pp. 585 587, 6. [5] Y. Zhang and B. E. Spielman, A stability analysis for timedomain method-of-moments analysis of -D double-negative transmission lines, IEEE Transactions on Microave Theory and Techniques, vol. 55, no. 9, pp. 887 898, 7. [6] G. L. Matthaei, L. Young, and E. M. T. Jones, Microave Filter, Impedance-Matching Netorks, and Coupling Structures,Artech House, Dedham, Mass, USA, 98.
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