CRLH Transmission Lines for Telecommunications : Fast and Effective Modeling

A different parameter extraction approach based on zero immittances for composite right/left-handed (CRLH) structure is presented. For lossless unit cell equivalent circuit model, LC parameters of series and parallel branches are extracted according to series resonance frequency and parallel resonance frequency, respectively. This approach can be applied to symmetric and unbalanced CRLH structures. The parameter extraction procedure is provided and validated by T-type unit cell model. The responses of distributed prototype and extracted equivalent LC circuit model are in good agreement. The equivalent circuit modeling can improve the degree of freedom in the CRLH TLs design. This parameter extraction method provides an effective and straightforward way in CRLH metamaterials design and applications in telecommunication systems.


Introduction
In telecommunication system, the role of the oscillator is to establish stable harmonic oscillation at a specific carrier frequency for modulation and mixing, which is shown in Figure 1(a).In the oscillator circuit, the main function of resonator is frequency selection and storage of energy.Conventional transmission line resonator is size dependent on frequency, which is not suitable for use in miniaturized integrated circuits.Composite right/left-handed transmission line (CRLH TL), introduced by Caloz and Itoh, is the transmission line (TL) approach of left-handed materials (LHMs) [1].CRLH metamaterials with right-handed (RH) and left-handed (LH) properties have the advantages of low loss and wide bandwidth, which have been extensively studied and broad applied in guided-wave and radiatedwave devices for over decades [2][3][4][5].Typically, the distributed CRLH TL is consisting of the series interdigital capacitor (IDC) and the shunt shorted stub inductor (SSI), which can be realized by the microstrip line and coplanar waveguide [4].Based on CRLH TL structures zeroth-order resonator can be achieved; its size is frequency independent and can be realized, theoretically, arbitrarily small [6].The schematic diagram of oscillator circuit using a zeroth-order resonator is shown in Figure 1(b).Using the zeroth-order resonator can effectively reduce the size of the oscillator circuit and improve the miniaturization of communication systems.
To accurately design CRLH TL and investigate the characteristics of CRLH TL-based devices, the equivalent circuit model is used to parameterize the complicated behaviors of the practical distributed circuits [7,8].The characteristics of the CRLH TLs are closely related to the LC parameters.Accurate design depends on whether the parameters meet the requirements.The design can be modified by parameter extraction in order to achieve optimum performances.In [9], to calculate the extracted LC parameters, the equivalent T and Π networks of IDC and SSI are separately characterized by admittance and impedance matrices; however coupling effects between IDC and SSI are not taken into account in this parameter extraction procedure.Otto et al. proposed an energy based method to extract equivalent circuit parameters [10], which was further implemented in [11].However, this method requires the use of software for electromagnetic energy calculation, cannot be extracted by experiment.Based on effective medium concept, revised Nicolson-Ross-Weir (NRW) approach was applied to extract equivalent parameters.In this method, the equivalent circuit model parameters are deduced by the effective permittivity   and the effective  permeability   .However, this method suffers multisolutions problem [12][13][14][15].The equivalent circuit extraction method proposed in [16] employs the transmission-line-reflect (TLR) calibration to deembed the effects of the transmission lines.
In this method both full-wave simulation and measured parameters are used to obtain the  transmission matrix of the CRLH unit cell.
In this paper, a different parameter extraction approach based on zero immittances has been presented.For lossless equivalent LC circuit model adopted, series resonance frequency and parallel resonance frequency are obtained from immittances characteristic, which is utilized to extract the LC parameters.In the proposed method, test transmission lines are placed in both sides of the CRLH structure.The effects of the transmission lines are eliminated by matrix calculation.The scattering parameter used can be obtained by full-wave simulations.The extraction procedure is convenient and effective and can be applied in unbalanced CRLH structures.This approach is illustrated by T-type unit cell model.The transmission characteristics of the distributed CRLH structure and the extraction LC equivalent circuit are in comparison with each other, and in wider bandwidth the extracted equivalent LC circuit model is effective.

Equivalent Circuit Modeling and Parameter Extraction Approach
The CRLH TL considered in this paper is constituted by series interdigital capacitor and parallel shorted stub inductors, which is shown in Figure 2(a).The dimensions are the same as those in [1], which is depicted in Figure 2 Based on the characteristics of immittances  se and  sh , we introduce an extraction approach to yield simple expressions for   ,   ,   , and   .For a two-port network, depicted in Figure 2(b),  se and  sh can be described in terms of  transmission matrix.Using standard network conversion formulas, the series impedance and parallel admittance can be written in terms of the  parameters, which lead to With zero immittances  se and  sh , the series resonance frequency and the parallel resonance frequency are determined by Im ( se )     se = 0, With  se and  sh , the LC parameters of series and parallel tank would be solved separately.And the corresponding extract parameters are determined by When  se =  sh , called balanced case, CRLH TLs would exhibit special properties.In practice, most initial fabrics are unbalanced, and with this extraction approach we can conveniently revise the design.That is, this extraction procedure is applicable for both balanced and unbalanced cases.

Extraction Procedure and Validation
This section provides the extraction procedure based on circuit model shown in Figure 3  Extracted parameter propagation phase.Therefore it is essential to eliminate the effect due to the test TLs.
The distributed CRLH TL and extracted equivalent circuit model are validated by comparing the scattering characteristics.The procedure is provided as follows: (1) Do full-wave simulation for the distributed circuit of Figure 5 and test microstrip TL to get the parameters of overall circuit and TLs, respectively.
(2) Compute [] of overall circuit and [] of TL corresponding to the -parameters obtained in (1) using standard network conversion formulas.
(3) Compute [] of CRLH TL using (4) Compute the  se and  sh vector using again standard network conversion formulas.
(5) Find zero points of  se and  sh using interpolation method to determine series resonance frequency  se and parallel resonance frequency  sh .
(6) Compute the   ,   ,   , and   using (4).The CRLH TL is balanced when the series and shunt resonant frequencies are equal; on the contrary, the CRLH TL is unbalanced when the series and shunt resonant frequencies are different.It is noted that the series and parallel LCs are extracted from  se and  sh , respectively.Regardless of whether the series resonant frequency and the parallel resonant frequency are equal, this extraction method can model the CRLH TLs.This makes it convenient to apply this approach in both balanced and unbalanced CRLH structures.

Conclusion
A zero immittances based parameter extraction approach for distributed CRLH structures has been proposed, which is applicable to symmetric unit cells.According to series resonance frequency  se and parallel resonance frequency  sh , determined by immittances characteristics, LCs of series branch and parallel branch are extracted, respectively, and therefore this method can be used in unbalanced CRLH structures.The presented extraction procedure provides an effective and straightforward way in CRLH metamaterials design and application of telecommunication system.Finally, the authors are currently applying this approach in analysis of CRLH metamaterials-based microwave circuit.

Figure 3 :
Figure 3: Equivalent circuit models of CRLH TL for parameters extraction.(a) T-type unit cell circuit model.(b) Equivalent network.

Figure 4 .
Figure 4.When we have zero immittances, the series resonance frequency  se and parallel resonance frequency  sh are obtained.
and verifies this method with CRLH TLs proposed by Caloz and Itoh.Usually, in order to obtain the transmission matrix of CRLH TL, a section of microstrip transmission line (TL) has to be added at each end of the component in full-wave simulation; the diagram is shown in Figure5.However, the added microstrip TLs would change the original network characteristics such as International Journal of Antennas and Propagation

Figure 6 :
Figure 6: Extracted LC parameters of equivalent LC circuit model.

( 7 )
With extraction parameters obtained from (6), simulate the equivalent LC circuit model and compare with distributed CRLH TL.The CRLH TLs proposed by Caloz and Itoh and shown in Figure 1 is considered here to illustrate the parameter extraction procedure with T-type unit cell circuit model.Fullwave simulation and numerical calculation are developed; the extraction results are presented in Figure 6.With different test TL lengths, the extracted LC parameters are mainly stable.It should be noted that, since the test transmission

Figure 7 :
Figure 7: Comparison of -parameters between distributed 1-cell CRLH TL and equivalent LC circuit model.