Issue 
EPJ Applied Metamaterials
Volume 3, 2016
MetamaterialbyDesign: Theory, Methods, and Applications



Article Number  8  
Number of page(s)  6  
DOI  https://doi.org/10.1051/epjam/2016006  
Published online  13 September 2016 
https://doi.org/10.1051/epjam/2016006
Research Article
Evenodd mode of a doubleLorentz metamaterial and its application to a triband branchline coupler
^{1}
Physics Department, Campus Rafic Hariri of SciencesLebanese University, Beirut, Lebanon
^{2}
IMEPLHAC, Grenoble INP, 03 Parvis Louis Néel, 38016
Grenoble, France
^{*} email: fatima.mazeh@live.com
Received:
30
March
2016
Accepted:
19
June
2016
Published online: 13 September 2016
The theoretical approach of a doubleLorentz (DL) transmission line (TL) metamaterial using evenodd mode analysis is presented for the application to a triband BranchLine Coupler (BLC). This BLC is based on doubleLorentz (DL) transmission line (TL) metamaterial to achieve the triband property. The triband operation is achieved by the flexibility in the phase response characteristic of such transmission line. Since metamaterials are in symmetric form, this analysis utilizes superposition and circuit symmetry to solve for the structure’s scattering parameters. A design example of a triple band quarter wavelength DL TL suitable for GSMUMTS applications is designed and evaluated by simulation using evenodd mode analysis to validate the proposed methodology at circuit level. Then, this simulated DL TL is used in the design of a triband BLC which is also being analyzed using evenodd mode analysis. This coupler exhibits transmission of 3 ± 0.5 dB, return losses and isolations larger than 14 dB, and a phase difference of ±90 ±3.5°.
Key words: Coupler / DoubleLorentz transmission line / Even mode / Metamaterial / Odd mode / Triband component
© F. Mazeh et al., Published by EDP Sciences, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Metamaterials (MTM) are artificial periodic structures with unusual electromagnetic properties fabricated with a negative effective dielectric permittivity and magnetic permeability. This corresponds to a new class named LeftHanded (LH) MTM which have gained significant interest in many guided waves and radiated applications. LH materials are so named because of the LH triad formed by the electric field, magnetic field, and wave vector leading to an antiparallel phase and group velocities [1]. Going through the transmission line (TL) approach, a LH TL is made up of periodic series capacitances and shunt inductances which is the dual of the conventional TL known as Righthanded (RH) TL. But a purely lefthanded (LH) TL doesn’t exist due to the natural parasitic induced current and voltage which are modeled by a series inductance and a shunt capacitance. This was the motivation for introducing the term CRLH (Composite RightLeft Handed) TL. The dual concept of such CRLH was introduced in [2]. However, the dual CRLH structure is an idealization that cannot be exactly recognized. A real dual CRLH MTM is in fact a doubleLorentz (DL) medium and this material has an intrinsic triband property that can be used to design various triband microwave components [3]. Both effective material parameters μ_{r} and ε_{r} of the corresponding line show Lorentztype dispersion.
Many microwave components are based on quarter wavelength transmission lines as BranchLine Couplers (BLC) [4]. But conventional quarter wavelength TLs known as RH TLs can operate only at their desired frequency and odd harmonics. Since wireless communication systems as GSMUMTS systems have operational nonharmonic frequencies, the conventional BLC can’t be an actual solution for them. Metamaterial (MTM) with its unusual properties helped to overcome many problems in the microwave world; one of which is increasing the number of operating frequencies. Triband components are helpful to reduce the size and the number of devices used in recent multiband telecommunication systems [5].
The natural BLC is modified by replacing the conventional transmission lines TLs known as righthanded transmission lines RH TLs with DoubleLorentz DL TLs to have a new one with three arbitrary operating frequencies. The advantage of using DL TLs over RH TLs is shown in the flexibility in the phase response diagram for which we can intercept a desired pair of phases at any arbitrary triple frequencies (f_{1}, f_{2}, f_{3}) for triband operation so that f_{2} and f_{3} are not necessary to be multiples of f_{1}. Triband components are helpful to reduce the size and the number of devices used in recent multiband telecommunication systems.
Two, three, or four port networks symmetric with respect to one or two planes are extremely implemented in RF and microwave devices. In [6], a full design of a DL TL was presented with useful design equations and an implementation of a triband branchline coupler was done using such type of MTM in [7]. Evenodd mode analysis is a classic topic for solving the scattering parameters of a symmetric circuit. A full analysis of symmetrical two port network and four port network is done in [8] and [9] respectively. The implementation of DL TL MTM using circuit models has been well investigated in the past few years but analyzed without taking the symmetrical advantage. However, calculations will be well simplified if a symmetric structure is divided into subcircuits. Since DL TLs MTM can be implemented using symmetrical model, one can analyze only half the circuit. In [10], an evenodd mode excitation is done for a bisymmetrical dualband BLC but not based on MTM. The main objective of this paper is to verify the use of evenodd mode analysis of metamaterial for a two port symmetric balanced structure of a DL TL to be extended in the use of a triband BLC which is a symmetrical four port network.
2 DoubleLorentz transmission line metamaterial
2.1 DoubleLorentz transmission line approach
The unit cell of the artificial DL TL consists of lumped elements L_{R} and C_{L} that are parallel in the series path and then of L_{L} and C_{R} that are series in the shunt path, a parasitic series inductance L_{P} and a shunt capacitance C_{P} as shown in Figure 1. A DL TL is designed by cascading periodically this unit cell with a condition that this cell is much smaller than the guided wavelength (λ_{g}) in the frequency range of operation. Mainly, it is examined in the homogeneous limit where (∆/λ_{g}) → 0.
Figure 1. Unitcell of artificial doubleLorentz (DL) transmission line (TL). 
As shown in Figure 1, the unit cell series impedance Z_{se} and shunt admittance Y_{sh} are given by (1) and (2):(1)where and .
The constitutive parameters μ_{eff} and ε_{eff} are plotted for a specific set of LC parameters in Figure 2.
Figure 2. DL TL metamaterial constitutive parameters for a specific set of LC parameters. 
The DL structure can be balanced so that no gap exists in the transition from LH medium to RH medium. There are two conditions to reach such case:(3)
Under the balanced condition, the dispersion relation and the characteristic impedance are given by (4) and (5):(4)
where .
2.2 Triband design procedure
The TL has six variables L_{P}, C_{P}, L_{R}, C_{R}, L_{L}, and C_{L} that should be calculated first. If we assume that the operating frequencies are chosen as f_{1}, f_{2}, and f_{3}, the phase shift of quarter wavelength DL TL at each frequency is given by (6)–(8):(6)
The phase shift is related to β by φ_{i} = −β_{i}N∆ where N is the number of unit cells and i = (1, 2, 3). So, the dispersion relation can be written in the form (9):(9)
2.3 Implementation
We noted that a DL TL is obtained by cascading the unit cell shown in Figure 1. However, to have equal input and output impedances, a balanced symmetric structure is recommended instead. A schematic of a symmetric DL TL with two unit cells is shown in Figure 3. Surface Mount Technology is used for the LH and RH components while the subsection that corresponds to the parasitic elements is implemented using microstrip lines.
Figure 3. Schematic of a symmetric DL TL with two unit cells. 
The procedure of implementation is summarized as follows:

Choose f_{1}, f_{2}, and f_{3}.

Solve the system of equations obtained in (9) for the unknown values of ω_{0}, ω_{∞}, and ω_{p}.

With the help of ω_{0}, ω_{∞}, ω_{p}, and Z_{0}, Calculate the values of L_{P}, C_{P}, L_{L}, C_{L}, L_{R}, and C_{R} which are derived to be:

Be sure that the operating frequencies are not found in the stopband in the dispersion diagram between righthanded media at lower frequencies and lefthanded media at higher ones. Otherwise, increase the number of unit cells chosen.

Use the values of L_{P} and C_{P} to find the lengths and widths of the microstrip lines using standard microstrip formulas.
3 Evenodd mode analysis of a DL TL
3.1 Symmetrical twoport network
A symmetrical network can be defined by a one having a plane of symmetry. Calculations will be well simplified when a two port network is divided into two structures mirroring each other [11]. This is a main requirement in analyzing complex symmetric structures. When an even excitation is applied to the network, the two applied signals at ports 1 and 2 are in phase. This creates a virtual open circuit symmetrical interface (“magnetic wall”). Similarly, under an odd excitation where the two applied signals are out of phase, the symmetrical interface is a virtual short circuit (“electric wall”) as shown in Figure 4.
Figure 4. Two port network (a) evenmode excitation (b) oddmode excitation. 
3.2 Scattering parameters
The network analysis will be simplified by analyzing each one port separately and then determining the twoport network parameters from the even and odd mode network parameters. The two port Sparameters are established where the subscripts “e” and “o” refer to the even mode and odd mode respectively [12]:(13)
3.3 Evenodd mode of a DL TL
A schematic of a symmetric 50 Ω DL TL using the procedure above is shown in Figure 5.
Figure 5. Schematic of a 50 Ω DL TL for: L_{1} = 16.25 mm, C_{R} = 0.55 pF, C_{L} = 7.24 pF, L_{R} = 1.205 nH, L_{L} = 18.24 nH. 
For even mode excitation, we can bisect the network with open circuits at the symmetrical interface as shown in Figure 6a. For odd mode excitation, we can bisect the network with short circuits at the symmetrical interface as shown in Figure 6b.
Figure 6. (a) Even mode excitation, (b) odd mode excitation. 
3.4 Simulation results
After bisecting the DL TL into two symmetric halves and applying the evenodd mode on the obtained two networks, simulation is done to find the S_{11} parameter for each one alone using [13]. However, the reflection coefficient S_{11} and the transmission coefficient S_{21} for the full DL TL can be directly obtained from (13) to (14) respectively and plot in Figure 7. The operating frequencies are 900 MHz, 1800 MHz, and 2100 MHz where the phase response is −90°, +90°, −90° respectively.
Figure 7. (a) Reflection coefficient of the DL TL using S_{11}even and S_{11}odd, (b) phase response S_{21} of the DL TL. 
4 Evenodd mode analysis of a BLC
4.1 Triband BLC
Following the previous procedure in Section 2.3, a BLC is implemented using 50 Ω and 35 Ω DL TLs using the schematic shown in Figure 5. The microstrip substrate used is FR4 with permittivity 4.4, thickness 0.8 mm, and copper thickness 18 μm. The operating frequencies are chosen to be 0.9 GHz, 1.8 GHz, and 2.1 GHz. The frequency dependence of the element components causes variations in the characteristic impedance of the DL TL, which results in an amplitude imbalance between the two output ports. To compensate this effect, a tuning stub is added to the 50 Ω DL TLs preserving the symmetric structure also. The length of the stub is tuned and found to be 2 mm. For more details, see [7].
4.2 Evenodd mode of a triband BLC
For a triband BLC, the structure will become more complex. To simplify calculation, let us consider the full symmetrical four port network with XX and YY symmetry axes. This is the case of a bisymmetrical structure where we can decompose the network into four single port subcircuits (eveneven, evenodd, oddeven, and oddodd) by the double application of the evenodd mode decomposition [14] as shown in Figure 8. The subscript 35 and 50 are used for the 35 Ω and 50 Ω TLs respectively.
Figure 8. Reduced subcircuits (a) eveneven (b) evenodd (c) oddeven (d) oddodd for L_{R}35 = 0.87 nH, C_{R}35 = 0.8 pF, L_{L}35 = 12 nH, C_{L}35 = 10.2 pF, L_{1}35 = 16.3 mm, W_{1}35 = 2.63 mm, L_{R}50 = 1.3 nH, C_{R}50 = 0.5 pF, L_{L}50 = 18.1 nH, C_{L}50 = 7.2 pF, W_{1}50 = 1.51 mm, L_{1}50 = 17 mm, L_{s} = 2 mm. 
The four port Sparameters are established as function of the single port networks parameters where the subscripts e and o refer to the even mode and odd mode respectively:(17)
4.3 Simulation results
After bisecting the BLC into four symmetric sections and applying equations (1) to (4), the simulated Sparameters are shown in Figure 9. The operating frequencies are 900 MHz, 1800 MHz, and 2100 MHz where the phase difference between the two output ports is ±90° ±3.5°. Figure 9 shows that the triband is well achieved where return losses as well as isolations are larger than 14 dB at each operating frequency; however, S_{21} and S_{31} are of −3 dB ± 0.5 dB.
Figure 9. Simulated Sparameters of the triband BLC using the single port subcircuit parameter. 
5 Conclusion
In this paper, an evenodd mode analysis of a DL TL metamaterial is presented. This DL TL has a triband property to be used in the design of triband microwave devices. So, this analysis was done to simplify calculations for complex circuits especially to those of periodic structure with much number of units as well as for complex structures used in EM simulators. This analysis has been also illustrated by a 50 Ω, λ/4 DL TL and a general description of any two port network is given first. Then, we extended our study to the application of a triband BLC using also evenodd mode analysis with bisymmetrical symmetry.
Acknowledgments
This work was partially supported by the Lebanese University (LU) and by the National Council for Scientific Research Lebanon (CNRS).
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Cite this article as: Mazeh F, Ayad H, Fadlallah M, Joumaa K, Jomaah J & Ndagijimana F: Evenodd mode of a doubleLorentz metamaterial and its application to a triband branchline coupler. EPJ Appl. Metamat. 2016, 3, 8.
All Figures
Figure 1. Unitcell of artificial doubleLorentz (DL) transmission line (TL). 

In the text 
Figure 2. DL TL metamaterial constitutive parameters for a specific set of LC parameters. 

In the text 
Figure 3. Schematic of a symmetric DL TL with two unit cells. 

In the text 
Figure 4. Two port network (a) evenmode excitation (b) oddmode excitation. 

In the text 
Figure 5. Schematic of a 50 Ω DL TL for: L_{1} = 16.25 mm, C_{R} = 0.55 pF, C_{L} = 7.24 pF, L_{R} = 1.205 nH, L_{L} = 18.24 nH. 

In the text 
Figure 6. (a) Even mode excitation, (b) odd mode excitation. 

In the text 
Figure 7. (a) Reflection coefficient of the DL TL using S_{11}even and S_{11}odd, (b) phase response S_{21} of the DL TL. 

In the text 
Figure 8. Reduced subcircuits (a) eveneven (b) evenodd (c) oddeven (d) oddodd for L_{R}35 = 0.87 nH, C_{R}35 = 0.8 pF, L_{L}35 = 12 nH, C_{L}35 = 10.2 pF, L_{1}35 = 16.3 mm, W_{1}35 = 2.63 mm, L_{R}50 = 1.3 nH, C_{R}50 = 0.5 pF, L_{L}50 = 18.1 nH, C_{L}50 = 7.2 pF, W_{1}50 = 1.51 mm, L_{1}50 = 17 mm, L_{s} = 2 mm. 

In the text 
Figure 9. Simulated Sparameters of the triband BLC using the single port subcircuit parameter. 

In the text 
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