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1、精品论文推荐Fast and accurate simulations of transmission-line metamaterials using transmission-matrix methodHui Feng Ma, Tie Jun Cui, Jessie Yao Chin, and Qiang ChengThe State Key Laboratory of Millimeter Waves, Department of Radio EngineeringSoutheast University, Nanjing 210096, P. R. China.Compiled Nov
2、ember 17, 2007Recently, two-dimensional (2D) periodically L and C loaded transmission-line (TL) networks have been applied to represent metamaterials. The commercial Agilents Advanced Design System (ADS) is a commonly-used tool to simulate the TL metamaterials. However, it takes a lot of time to set
3、 up the TL network and perform numerical simulations using ADS, making the metamaterial analysis inefficient, especially for large-scale TL networks. In this paper, we propose transmission-matrix method (TMM) to simulate and analyze the TL- network metamaterials efficiently. Compared to the ADS comm
4、ercial software, TMM provides nearly the same simulation results for the same networks. However, the model-process and simulation time has been greatly reduced. The proposed TMM can serve as an efficient tool to study the TL-network metamaterials.51. IntroductionIn 1968, Veselago first proposed the
5、concept of left- handed material (LHM) 1. The new medium exhibits many particular properties because of its negative per- mittivity and negative permeability. However, LHM does not exist in nature, which has to be realized using peri- odic structures with the unit cell of split-ring resonators (SRR)
6、 and conducting wires 2-5 or electric LC (ELC) resonators 6. The two-dimensional (2D) L and C loaded transmission-line (TL) networks is another method to re- alize LHM 7-17. Both isotropic 7-14 and anisotropic TL metamaterials 15-17 have been studied. The partic- ular 2D L and C loaded TL networks c
7、an be equivalent to relevant media. In order to obtain the voltage and current distributions on the periodically TL networks, which are equivalent to the field distributions in meta- material, the Agilents Advanced Design System (ADS) has been used 8-17. However, it took much time and vigor to set u
8、p the LC-loaded TL networks when using ADS, especially for large-scale structures. Also, the nu- merical simulations were time consuming for large-scale structures using ADS.Are there some methods which can be used to set up the model quickly and to simulate the TL strucutures efficiently? The trans
9、mission-matrix method (TMM) is a good choice 18. Caloz and Itoh have studied such a method in Ref. 19 to analyze TL metamaterials. How- ever, they only considered the periodic structures com- posed of lumped inductance and capacitance, the com- posed right/left-handed circuits, in which the transmis
10、-tion nodes.In this paper, we generalize the TMM approach to study the 2D LC-loaded TL periodic structures, which have been widely used to represent metamaterials 7-17. Both the voltage and current distributions on the LC- loaded TL structures can be computed accurately and efficiently using the pro
11、posed TMM, which have excel- lent agreements to the ADS simulation results. However, the time used in the model setup and numerical simu- lations have been greatly reduced. The proposed TMM approach can also solve large-scale TL metamaterials, which can hardly be realized using ADS.2. Theoretical An
12、alysis of TMMThe arbitrary 2D TL network under consideration is di- vided into m n unit cells, as shown in Fig. 1(a). The corresponding general T structure of each unit cell is illustrated in Fig. 1(b), in which Z0 , and l are the characteristic impedance, the propagation constant, and the length of
13、 TL section, respectively; and Zs and Yp are loaded inductance and capacitance. Using TMM, we di-vide the structure into columns of unit cells, one of which is shown as the gray line in Fig. 1(a). The m 1 unit cells can be represented as a 2m 2m matrix formed by m input ports and m output ports with
14、 the node voltages and port currents 19. The equivalent column cell with terminated impedance Rj is shown in Fig. 2. The input and output ports are indexed as 1 to m and m + 1 to2m, respectively, and the corresponding 2m 2m matrixis expressed assion lines are not involved 19. Also, the voltage and c
15、ur- rent distributions obtained from this method cannot be equivalent to those from the ADS simulations, because Vin Iin Vout = Tj Iout ,(1)the load impedances of observers are added to observa-which can be referred to Eq. (4.18) in Ref. 19.Z s 2Z s 2Z s 2l 2 Z s 2Z 0Y p(a) (b)Fig. 1. (a) Arbitrary
16、2D L and C loaded TL network. (b) General T network of each unit cell.Fig. 3. The whole network under consideration, including m inputs and m outputs, which is divided into the input network Tin , the center network Tc , and the output network Tout . A source is arbitrarily located at the kth row an
17、d between the ith and (i + 1)th columns of the network.Z s 2l 21V Z s 2I1R jZ 0l 2 Z s 2Z 0YpVm 1I m 1Th1V1 Z s 21I l 2Z 0R jl 2 l 2Z s 2Tv Z 0ZZ0YpTh2 Z s 2 Vml 2Im0sion matrix as shown in Fig. 1(a) will be determined asnT = Y Tj .(3)j=1Z s 2Z s 23. Voltage and Current DistributionsWe consider the
18、case of an m n network with a givenZ s 2l 22V Z s 2Z 0l 2 Z s 2Vm 2l 2 l 2Z s 2V2Z s 2Z 0 Z s 2 Vm 2source which is arbitrarily loacted at the kth row and be- tween the ith and (i + 1)th columns of the network. WeYI 2 Z 0pZ s 2I m 2I l 22Z 0Z 0Z s 2l 2Yp Z 0I m 2determine to obtain the output voltag
19、e and current atany point of the network. The whole network is demon-strated in Fig. 3, where R1 , R2 , , and Rm are hori- zontally terminated impedances. LetZ s 2l 2Z s 2VmZ 0l 2 Z s 2V2 mZ s 2Vml 2 l 2Z s 2Z 0Z s 2 V2mTin = Tin TRA =inBin ,(4)Z 0I m Y pZ s 2I 2 mI ml 2Z 0Z 0Z s 2l 2Yp Z 0I 2mCinDi
20、nR j R jTout = Tc Tout TR = AoutBo ut ,(5)Fig. 2. The equivalent-circuit T network for a column cell and the corresponding decomposition. (a) T network for a column of cell. (b) The decomposition of three parts.in whichandTR =Cout Dout IX ,(6) O I R1 0 0By decomposing the column cell into three sub-
21、columncells, as shown in Fig. 2, we can easily obtain the 2m2mX = 0R2 0 .(7)transmission matrix Tj as the product of three matricesT h1, T v and T h2:. . . .000RmIn Eqs. (4) and (5), A, B, C, D, A , A B= T h1 T T h2 ,(2) in ininin outTj =C D vBout , Cout , Dout and X are all m m matrices. Using the
22、series transmission-matrix theory, we caneasily obtain the following equationswhere the computation of matrices T h1, T v and T h2is similar to that in Ref. 19.Once Tj has been obtained, the n-column transmis- Vin I11 = Tin O I ,(8) Vin I12 = Tout O hI i#,(9)Comparing with the ADS simulations 9, we
23、notice that two results have excellent agreements in both amplitude and phase distributions. The results show that it is validhere, O is m1 zero matrix, I and hI i are the currentas illustrates in Fig. 3. From Eqs. (8) and (9), we have 1 to replace ADS by TMM in numerical simulations of TLmetamateri
24、al structures. In this example, TMM does not exhibit its advantage obviously in saving time with thesmall-scale network.I11 = Din Bin Vin = Yin Vin ,(10)To verify the accuracy of TMM further, we give a full comparison of TMM-simulation results to the commer-1 I12 = Dout Bout and Vin = Yout Vin ,(11)
25、cial ADS-simulation results. We consider an example ofsuper waveguide, which is a planar waveguide filled with air and left-handed material (LHM) 14,20. It has beenIin = I11 + I12 .(12) Furthermore, we obtain1shown that extremely high-power densities with opposite propagation directions can be gener
26、ated and transmit- ted along the waveguide if the air and LHM have equalthickness and the permittivity and permeability of LHMVin = (Yin + Yout ) Iin = Y Iin .(13)According to Eq. (4.32) in Ref. 19, the input currentIin m1 will be zero except the kth-row current Iin,k . 18The kth-row input voltage V
27、in,k = Vs Rs Iin,k . Then 16we have 1412RowVs10Source1st Interface2nd Interface1510Voltage (dB)5Strong Evanescent 0FieldsIin,k =Rs+ ykk,(14)86InternalExternal510in which ykk is the kth-row and kth-column element ofx 4matrix Y . 2FocusFocus15205 10 15 20 25We remark that the kth element of the matrix
28、 Iin is Iin,k , and all other elements are zero. Hence, once weobtain Iin,k , the matrix Iin will be known, and thenmatrices Vin , I11 and I12 will also be determined fromEqs. (13), (10) and (11), respectively.Once we know Vin , I11 and I12 , the output voltagezVoltage Along Row 10 (dB)2010010TL Mes
29、hSourceDual TL LensInternalFocusTL MeshStrong EvanescentFieldsExternalFocusand current at any point of the network which is locatedon the right side of the source will be determined easily205 10 15 2025Columnfrom the following equationFig. 4. TMM-simulation results for TL super lens 9 at 1GHz (volta
30、ge amplitude).1inC Vout = TIout V I12 .(15)Similarly, we can obtain the output voltages and cur-SourceInternalFocusExternalFocus5 1015Column20 2518rents at points located on the left side of the source.Using the same method, we can also deal with the cases 16of two given sources which are located at
31、 the same row. 1410The above method can be used to analyze the elec-12Rowtromagnetic properties of 2D TL metamaterials such as super lens 7-11, EM localizations 12,13, and super8waveguides 14 with high efficiencies and high accura-cies. 6244. Simulation ResultsIn order to verify the correctness of t
32、he proposed TMM method, we re-compute the sub-wavelength focusing structure possessing the same TL network and parame-1st interface2nd interface050100150Phase (degrees)200250300350ters as those in Ref. 9 using TMM. The correspond- ing TMM-simulation results are shown in Figs. 4 and 5.Fig. 5. TMM-sim
33、ulation results for TL super lens 9 at 1GHz (voltage phase).satisfy = 0 (1 + ) and = 0 /(1 + ), in which is a small parameter 20. The above theoretical prediction has been realized using 2D LC-loaded TL metamateri-als 14. Now we re-analysis the TL-metamaterial super waveguide using the TMM technique
34、.Consider a TL-metamaterial super waveguide whichis composed of 18 179 unit cells 14, where the right- handed TL (RHTL) and the left-handed TL (LHTL) parts extend 9 cells in the x direction and 179 cells in they direction, respectively. A 1-V (0 dB) voltage source isconnected to the node of the cell
35、 numbered as (4, 90) inthe RHTL region. We choose the size of unit cell as d = 1 cm, the small parameter as = 0.053, the characteris- tic impedance in TL as Z0 = 533.1459 , and the prop- agation constant in TL as = 14.8096. The simulation results of voltage and current distributions computed byFig.
36、8. The comparison of voltage and current distributions on the planar TL-metamaterial super waveguide along the line y = 90, where red lines indicate the ADS simulation results, and the dashed blue lines are TMM computation re- sults. (a) The amplitude of voltage. (b) The phase of voltage. (c) The am
37、plitude of current. (d) The phase of current.Fig. 6. The voltage and current distributions on the planar TL-metamaterial super waveguide with 18 179 unit cells computed by TMM. (a) The amplitude distribution of volt- age. (b) The phase distribution of voltage. (c) The amplitude distribution of curre
38、nt. (d) The phase distribution of current.Fig. 7. The voltage and current distributions on the planar TL-metamaterial super waveguide with 18 179 unit cells simulated by the ADS commercial software. (a) The ampli- tude distribution of voltage. (b) The phase distribution of voltage.TMM are illustrate
39、d in Fig. 6. As a comparison, the cor- responding voltage distributions simulated by ADS are shown in Fig. 7. From these two figures, excellent agree- ments have been observed for the voltage distributions. The current distributions have also excellent agreements, which are not repeated here.We rema
40、rk that much less time has been used in TMMthan that in ADS, since it takes long time to label the voltage and current probes in ADS for the 18 179 net- work. At least several days were taken to set up and simulate such a large-scale TL metamaterial using ADS, while it only takes several minutes to
41、solve the same problem using TMM.To further verify the accuracy of TMM, the voltage and current distributions on the planar TL-metamaterial super waveguide along the x direction with y = 90 are computed using TMM and ADS, as shown in Fig. 8. Clearly, the two results have excellent agreements.Then we
