Analyzing of Coupling Region for CRLH/RH TL Coupler with Lumped-elements

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1 PIERS ONLINE, VOL. 3, NO. 5, Analyzing of Coupling Region for CRLH/RH TL Coupler with Lumped-elements Y. Wang 2, Y. Zhang, 2, and F. Liu 2 Pohl Institute of Solid State Physics, Tongji University, Shanghai 292, China 2 School of Electronics and Information, Tongji University, Shanghai 292, China Abstract The coupling region of the CRLH/RH TL coupler, which is composed of a composite right/left-handed transmission line (CRLH TL) with lumped-elements and a conventional right-handed transmission line (RH TL), is explicitly decided. Detailed formulas are given to define the frequency band edge of the coupling region. ADS S-parameter simulations are demonstrated to confirm our theoretical results. DOI:.2529/PIERS INTRODUCTION Over the last years, left-handed materials based on transmission line approach have become very popular due to their potential application in microwave community. This approach to realize metamaterials was first introduced at approximately the same time by three different groups [ 3]. Caloz and Itoh have referred to this type of left-handed materials as composite right-/left-handed transmission lines (CRLH TL). Recently, CRLH TL have led to the development of several new component and devices, such as a leaky-wave backward antennas [4], phase-shifting lines [5] and couplers [6 8]. The asymmetric CRLH/RH TL coupler was first introduced and studied in [7] and [8]. It was composed of a conventional right-handed transmission line (RH TL) and a composite right/lefthanded transmission line (CRLH TL), which was realized by loading a conventional microstrip line with series capacitors and shunt inductors. The coupler showed superior performance such as broad bandwidth and tight coupling characteristics. The interesting features of such asymmetric CRLH/RH TL coupler were discussed using coupled-mode theory based on traveling waves. In Ref [9], it was shown that the formation of a stop-band and the excitation of complex modes occurred in the case of coupling between a forward wave and a backward-wave mode for a range of frequencies around the tuning frequency. Moreover, Ref [] showed that by adjusting the spacing between the RH TL and CRLH TL, backward coupling operates in the left-handed region. In this paper the coupling region of the asymmetric CRLH/RH TL coupler is intensively studied. The conditions for tight coupling are presented and detailed formulas are given to define the edges of the coupling region. d / 2 d / 2 2 C L Z L L 2 C L Figure : Circuit model for a unit cell of the CRLH TL, L L and C L are loaded lumped-elements. 2. CHARACTERISTICS OF A CRLH TL A CRLH TL has been theoretically investigated in [4] and [6]. It can be implemented by loading a conventional transmission line with lumped element series capacitors (C L ) and shunt inductors (L L ) as depicted in Fig.. In this structure, the loading elements represent the left-handed attributes

2 PIERS ONLINE, VOL. 3, NO. 5, and the interconnecting transmission provides the right-handed contribution. Fig. 2 shows the dispersion diagram of the CRLH TL, which illustrates that it has a left-handed region and a righthanded region. The left-handed region has double-negative parameters, where both the permittivity and permeability are negative. Meanwhile, the right-handed region behaves as a conventional transmission line with both positive permittivity and permeability. It is also shown that a CRLH TL whose series and shunt resonant frequencies are not equal (f se f sh ) displays a stopband Frequency (GHz) right-handed region stopband left-handed region max( f se, f sh ) min( f se, f sh ).5 f cutoff β d (rad) Figure 2: Dispersion diagram of the CRLH TL. For a CRLH TL, the cutoff frequency of the left-handed region is given by f c = 4π L L C L () and the effective permittivity and permeability are given by the following approximate expressions ε = ε(ω) = ( C R ) p ω 2 (2) L L d ( µ = µ(ω) = p L R ) ω 2 (3) C L d where, p is the structural constant, L R (H/m) and C R (F/m) are the distributed series inductance and shunt capacitance of the interconnecting transmission line. C L (F) and L L (H) are the parameter of lumped elements. The parameter d is the length of unit-cell. From Eq. (2), it can be shown that for µ <, the frequency satisfies f < 2π L R C L d = f se (4) Similarly, from Eq. (3), it can be shown that for ε <, the frequency satisfies f < 2π L L C R d = f sh (5) Therefore, the left-handed passband where µ < and ε <, is given by f c < f < min(f se, f sh ). 3. DISCUSSION ABOUT THE COUPLING REGION 3.. ENG CRLH TL/RH Coupler A twenty-unit CRLH TL with circuit model in Fig. is designed. The width of the transmission line is mm with FR-4 substrate of ε r = 4.75, of which the distributed series inductance L R and shunt capacitance C R equal to nh/m and pf/m respectively. The loading series capacitors are C L = 5. pf and shunt inductors L L = 4.7 nh. The unit cell dimension is d = 7 mm. The cutoff frequency defined by Eq. () is f c =.5 GHz. The series and shunt resonant frequencies

3 PIERS ONLINE, VOL. 3, NO. 5, d/2 d/ C L 2 C L L L Figure 3: Circuit model for a CRLH/RH TL coupler, L L and C L are loaded lumped-elements. defined by Eqs. (4) and (5) can be obtained as f se =.49 GHz and f sh = 2.45GHz. Therefore, the left-handed passband of this CRLH TL is f c < f < f se. We refer this CRLH TL as ENG CRLH TL for the stopband of this CRLH TL is epsilon-negative. When a conventional right-handed transmission line of the same length and width is near this CRLH TL, coupling will occur between the two lines. Then, the two lines constitute a CRLH TL/RH TL coupler with circuit model in Fig. 3. In fact, the coupling occurs only between the conventional transmission line and the part of interconnecting transmission line in the CRLH TL, but not between the transmission line and lumped elements in CRLH TL. As a result, the distributed series inductance L R and shunt capacitance C R of the interconnecting transmission line will be modified correspondingly, which will then lead to the changes of f se and f sh. However, the cutoff frequency f c keeps constant because coupling won t change the values of C L and L L. When the space between the two lines is.2 mm, the distributed series inductance and shunt capacitance will change to C R = 4.8 pf/m and L R = 26. nh/m []. Then, the edges of the stopband become f se =.62 GHz and f sh = 2.33 GHz. Therefore, the left-handed region becomes f c < f < f se. Figure 4 shows the change of the through coupling parameters for the CRLH TL when the RH TL is put nearer and nearer. No coupling occurs if the original spacing between the two lines is infinity. The edges of the stopband are shown to be.49 GHz and 2.45 GHz. During the decreasing of the spacing, the bandgap edges move to.65 GHz and 2.4 GHz which confirm our theoretical prediction. What s more, a new gap appears on during decreasing of the spacing between the two lines. This gap appears because of the coupling, so this frequency band corresponds to the coupling region. It demonstrates coupling region is just in the left-handed region of the CRLH TL. Therefore, the coupling region must satisfy f c < f < f se frequency(ghz).6mm.8mm.2mm Figure 4: of the ENG CRLH TL when the spacing between the lines is changing from.2 mm to infinity large µ rr = +ε rr = ε µ < Permittivity Permeability ε rr -µ rr Figure 5: Relative premittivity and permeability of the ENG CRLH TL and relative premittivity ε rr and permeability µ rr of the RH TL. Figure 5 shows the and of the CRLH TL and the negative µ rr and ε rr of the RH TL. The effective permeability and permittivity of the CRLH TL are defined by Eqs. (2) and (3). However, µ rr and of the RH TL are constant and equals to. and 3.56 respectively. It can be seen that µ rr = (i.e., µ = ) at f µ= =.6 GHz, and ε rr = (i. e., ε = ) at f ε= =.74 GHz. The frequency band with ε µ <. Therefore, the Coupling occurs at the

4 PIERS ONLINE, VOL. 3, NO. 5, frequency band where the average permittivity and permeability satisfy ε µ <. Therefore, the coupling region must satisfy f µ= < f < f ε= for ENG CRLH TL/RH TL coupler. For an asymmetric ENG CRLH/RH TL coupler, the coupling region satisfies ε µ < and <, < at the same time, therefore, the coupling region is f µ= =.6 GHz < f < f se =.62 GHz. Fig. 6 shows that coupling region is from.5 GHz to.65 GHz and confirms our theoretical prediction MNG CRLH TL/RH TL Coupler If the stopband of the CRLH TL is mu-negative, the stopband is f sh < f < f se and the left-handed region is f c < f < f sh. For an asymmetric MNG CRLH TL/RH TL coupler, the coupling region satisfy f ε= < f < f sh. Another MNG CRLH TL is designed. The width of the interconnecting transmission line is still mm. The length of the unit is 7 mm, The loading series capacitors are C L = pf and shunt inductors L L = 4.7 nh. The gap edges are derived to be f se = 3.36 GHz and f sh = 2.45 GHz. The cutoff frequency is f c =.6 GHz. - - S-parameters S S 3 S mm.8mm.2mm Figure 6: Simulating S-parameters for the CRLH/RH TL coupler. Figure 7: of the MNG CRLH TL when the spacing between the lines is changing from.2 mm to infinity large. As same as the case of ENG, when a conventional right-handed transmission line is near this CRLH TL, coupling will influence the edges of the bandgap. As we have mentioned above, the distributed series inductance and shunt capacitance of the CRLH TL will change to C R = 4.8 pf/m and L R = 26. nh/m when the spacing between the two lines is.2 mm. Therefore, the gap edges become f se = 2π L R C = 3.67 GHz Ld f sh = 2π C R L = 2.33 GHz Ld Figure 7 shows f se moves to 3.73 GHz and f sh moves to 2.25 GHz. In this case, the left-handed passband is f c < f < f sh for MNG CRLH TL. The permeability and permittivity of the CRLH TL and the negative µ rr and ε rr of the RH TL are shown in Fig. 8. It can be seen that ε rr = at f ε= =.67 GHz and µ rr = at f µ= = 2.38 GHz. The coupling S-parameter of a MNG CRLH TL/RH TL coupler when the space is.2 mm is demonstrated in Fig. 9. It shows the coupling region from.67 GHz to 2.25 GHz, satisfies f ε= < f < f sh. 4. CONCLUSION The coupling region of the CRLH/RH TL coupler is studied. The coupling region of the CRLH/RH TL coupler satisfy ε µ < and < and < at the same time. When the coupling occurs between an ENG CRLH TL and a RH TL, the lower edge of the coupling region defined by

5 PIERS ONLINE, VOL. 3, NO. 5, ε rr = +µ rr = ε µ < Permittivity Permeability -µ rr -ε rr S-parameters S S 3 S 4 Figure 8: Relative premittivity and permeability of the MNG CRLH TL and relative premittivity ε rr and permeability µ rr of the RH TL. Figure 9: Simulating S-parameters for the MNG CRLH/RHTL coupler. + µ rr =. The higher edge of the coupling region is f se. The stop band of the CRLH TL becomes f se < f < f sh. When the coupling occurs between a MNG CRLH TL and a RH TL, the lower edge of the coupling region defined by + ε rr =. The higher edge of the coupling region is f sh. The stop band of the CRLH TL becomes f sh < f < f se. ACKNOWLEDGMENT This research was supported by National Basic program (973) of China (No. 2CB646 and No. 24CB7982) and by the National Natural Science Foundation of China (No and No ). REFERENCES. Iyer, A. K. and G. V. Eleftheriades, Negative refractive index matematerials supporting 2-D wave, IEEE MTT-S Int. Microwave Symp. Dig., 67 7, Caloz, C. and T. Itoh, Application of the transmission line theory of left-handed(lh) materials to the realization of a microstrip LH Line, IEEE AP- S Int. Symp. Dig., Vol. 2, 42 45, Oliner, A. A., A periodic-structure negative-refractive-index medium without resonant elements, IEEE AP-S/URSI Int. Symp. Dig., 4, Grbic, A. and G. V. Eleftheriades, Experimental verification of backward-wave radiation from a negative index metamaterial, J. Appl. Phys., Vol. 92, No., , Nov Antoniades, M. and G. V. Eleftheriades, Compact, linear, lead/lag metamaterial phase shifters for broadband applications, IEEE Antennas Wireless Propag. Lett., Vol. 2, No. 7, 3 6, Caloz, C., A. Sanada, and T. Itoh, A novel composite right/left-handed coupled-line directional coupler with arbitrary coupling level and broad bandwidth, IEEE Trans. Micro. Theo. Tech., Vol. 52, , Caloz, C. and T. Itoh, A novel mixed conventional microstrip and composite right/left-handed backward-wave directional coupler with broadband and tight coupling characteristics, IEEE Micro. Wireless Comp. Lett., Vol. 4, No., 3 33, Islam, R. and G. V. Eleftheriades, A planar metamaterial co-directional coupler that couples power backward, IEEE-MTT Int. Symp. Dig., , June Islam, R. and G. V. Eleftheriades, Printed high-directivity metamaterial MS/NRI coupledline coupler for signal monitoring applications, IEEE Micro. Wireless Comp. Lett., Vol. 6, No. 4, 64 66, April 26.. Wang, Y., Y. Zhang, L. He, F. Liu, H. Li, and H. Chen, Discussion on coupling mechanism of asymmetric CRLH/RH TL coupler, J. Zhejiang Univ. Science A, 95 98, 26.. Kirschning, M. and R. H. Jansen, Accurate wide-range design equations for the frequencydependent characteristic of parallel coupled microstrip lines, IEEE Trans. Micro. Theo. Tech., Vol. MTT-32, 83 9, Jan. 984.