Analysis of Characteristics of Coplanar Waveguide with Finite Ground-planes by the Method of Lines

Transcription

1 PIERS ONLINE, VOL. 6, NO. 1, Analysis of Characteristics of Coplanar Waveguide with Finite Ground-planes by the Method of Lines Min Wang, Bo Gao, Yu Tian, and Ling Tong College of Automation Engineering, University of Electronic Science and Technology of China Chengdu 614, China Abstract The method of lines MoL has been used to analyze the characteristics of various structures of coplanar waveguide widely. However, a large number of literatures do not study the effects of the ground-plane width on the propagation characteristics, especially characteristic impedance. In this paper, the characteristics effective dielectric constant and especially characteristic impedance of shielding coplanar waveguide with finite ground-planes are analyzed by MoL and the computation results are shown. 1. INTRODUCTION Presently, planar transmission lines have predominantly been used in microwave integrated circuits MICs as well as monolithic microwave integrated circuits MMICs. And coplanar waveguide CPW is considered more suitable for MIC and MICC applications than conventional microstrip line because the CPW offer several advantages over conventional microstrip line: First, it simplifies fabrication; second, it facilitates easy shunt as well as series surface mounting of active and passive devices 1; third, it eliminates the need for wraparound and via holes 1, and fourth, it reduces radiation loss 1. Since the adoption of CPW in certain MIC and MMIC applications, the need for an in-depth analysis of this structure has increased. As is well known, the reason for making the lateral ground planes of finite extent is because this more closely models a practical CPW circuit. The finite-difference time-domain FDTD method 2 has been used to analyze a given CPW configuration, but it would take lots of time. So the method of line MoL 3 is an effective method to analyze the CPW to clarity the impact of the change of ground-plane widths on dispersion and attenuation characteristics. In this paper, MoL with nonequadistant discretization to analyze the characteristics of CPW with finite ground-planes is presented. The shielding CPW as an example, the propagation constants are calculated with the frequency up to GHz. The influence of finite ground planes is focus to be discussed. 2. ANALYSIS PROCEDURE Figure 1 illustrates the structure under investigation. Layer II is the lossy substrate,and groundplanes as well as center-strip are assumed to be prefect electric conductors PECs on the substrate. Layer I and layer III are the vacuum below and above the substrate. The top, bottom and lateral boundaries are also PECs, so it do not include raditation effects. Thus, the resulting attenuation is only owing to substrate loss. Due to symmetry, only half of the structure needs to be analyzed with a magnetric wall in the center for simplifying calculation. We use the MoL to analyze this structure, and we start with the independent field components e z and h z. From Maxwell s equations, we find that e z and h z must fulfill the Helmholtz equation 2 ψ x ψ y ψ z 2 + k2 ψ Lψ k 2 ε r k 2 ω µ ε 1 in each separate layer for ψ we have to substitute either e z or h z. Moreover, e z and h z must fulfill the following boundary conditions: Electric Wall Magnetic Wall e z D; h z D; h z n N 2 e z n N

2 PIERS ONLINE, VOL. 6, NO. 1, Figure 1: Cross section of shielding coplanar waveguide with finite ground-planes. The fields, the field equations, and the wave equations Helmholtz equations for E z and H z of each layer are discretized in one direction along the x coordinate in Fig. 1. The steps of nonequidistant discretization are studied in detail in 3. And the discretized differential quotients can be written in the following way: hr 1 h diag ez r h DE z r h Dr e E nz DE nz x i hr 1 hz e diag r e D t H z r e D t r h H nz D t H nz x i h 2 r 1 2 e 3 z i e diag x 2 D t DE nz P DN E nz h 2 r 1 2 h diag h z i x 2 D D t H nz P ND H nz with r e diag h/e i and r h diag h/h i. The difference operators P DN and P ND are real symmetric tridiagonal matrices. Assuming E z or H z wave propagation in z direction according to e jkzz, thus we obtain with Eq. 3 the ordinary differential equation which can be written as d 2 dy 2 + k2 kz 2 I h 2 λ 2 Ψ n 4 with Ψ n T Ψ n and T PT λ 2, where T is the transformation matrices for a homogeneous dielectric layer and the eigenvector matrix belonging to P. λ 2 is the eigenvalue matrix. Let 2 k 2 λ2 i ε r + ε re, ε re k2 z k 2, λ2 i λ2 i h 2 Since in most cases the components and their derivatives are only needed on the layer interfaces, we give the solution for a homogeneous dielectric layer with thickness d see Fig. 1 also in the following form at the interfaces A and B: Ψ A k 2 γ α Ψ A y 6 α γ Ψ B Ψ B with Ψ 1 d dy Ψ k y diag 1 α diag sinh k yi d 1 7 γ diag tanh k yi d

3 PIERS ONLINE, VOL. 6, NO. 1, After some algebraic manipulations using Eq. 6, the following system equations for the tangential field components is obtained and can be written in a shorter way detailed in 3: H A ȳ1 ȳ 2 ĒA 8 H B ȳ 2 ȳ 1 Ē B where Ē A,B Ē xa,b jē za,b j H H A,B η za,b H xa,b In layer I and III, the tangential components of the electric field are zero at the interface top and bottom. According to Eq. 8, H A ȳ I 1ĒA In layer II, according to Eq. 1, we can obtain H B ȳ II 2 9 H B ȳ1 III Ē B 1 ȳi 1 + ȳ1 II 1 ȳii 2 ȳ1 II ĒB 11 In the next step we have to take care of fulfilling the continuity conditions at all the interfaces. At this interface the matching equations hold Ē II B Ē III B Ē B H II B H III B J B From Eq. 11 and Eq., we can obtain Th Z 11 Z T t h jjxb T e Z 21 Z T t e J zb Z11 Z Z 21 jjxb Z J zb E xb je zb For interface B in Fig. 1, the electric field E on the metallization in the slot and the electric current J in the slot are zero. So we rewrite Eq. 13 in a conventient way. In the first step we omit the columns mentioned in each submatrix Z ik i, k 1, 2, yielding the so-called reduced matrix. In the second step we partition each of the rectangular submatrices in an upper superscript u and a lower superscript l submatrix according to the partition of the vector on right-hand side, and we obtain two system equations Z rl 11 Z rl and Z ru Z rl 21 Z rl 11 Z ru Z ru 21 Z ru jjxm J zm jjxm J zm E xs je zs The superscript r means reduced in connection with the first step. The subscript s in E x and E z stands for slot and the subscript m in J x and J z for metallization. Because J xm and J zm, the nontrivial solution of Eq. 14 requires Zrl 11 Z rl 16 Z rl 21 Z rl Using the numerical method, such as the Newton iteration method and muller method, the normalized propagation constant ε re and attenuation constant α can be obtained. In waveguides where hybrid waves can propagate, characteristic impedances Z are defined by means of the power transfer P. For example, in this symmetrical coplanar waveguide it is defined as Z U 2 /P where U is the integral of the electric field from one finite ground-plane edge in the slot to the center strip. P is determined as the integral of the Poynting vector over the waveguide cross section F. And P must be calculated for each layer of the planar structure separately and summed over all layers. For the layer II between interface A and y 2 interface B in Fig. 1, we obtain y2 P e x h y e y h x dxdy h E t x H y E t y2 Ēt yh x dy h x Hy Ēt H y x d7 F

4 because PIERS ONLINE, VOL. 6, NO. 1, E t xh y Ēt xt t T H y Ēt x H 8 When we get the second system Eq. 1, the current vector jj t xm, J t zm t is determined as an eigenvector afterwards. If ε re and the current vector are evaluated, the field vector E t xs, je t zs t can be calculated with the system Eq. 1 and all the other field components in the various interfaces can be obtained. Then we can write for the first term in Eq. 17 y2 Ē t H x y dy 1 t ĒxA g1 g 2 HyA 19 2 Ē xb g 2 g 1 H yb where g 1 k 1 γ h dk 2 yh α2 h g 2 k 1 k dk 2 h α hγ h α h and α, γ and k y are according to Eq. 7. From these steps, P of the layer II can be obtained, and P of the other layers should be obtained as the same steps. We also get the voltage U from the electric field from either ground-plane to the center conductor and then we obtain Z. 3. NUMERICAL RESULTS In this Section, numerical results will be presented to assess the results of the computations as well as to investigate the effect of finite ground-planes on the propagation characteristics of shielding CPW structure. To verify the validity of this method, the frequency-dependent ε re and Z computed by MoL are compared with that calculated by HFSS and the results in 4. Fig. 2 shows that ε re computed by MoL agrees well with that by HFSS and is slightly larger than ε re in 4. This is because the ground planes in 4 extend to lateral board boundaries. From Fig. 3, it can be observed that the values of ReZ and ImZ calculated by both MoL and HFSS are close to each other. It is worth mentioning that ReZ first increases slowly with frequency, reaches its peak at around 6 GHz, and then decreases rapidly. Figure 4 shows the values of ε re, α, ReZ and ImZ calculated separately with different finite ground-planes width g:.6 mm, 1 mm and 9 mm. The values of ε re and α increase with the increasing frequency. Three curves of α with different g almost overlap. It s because finite ground planes can hardly influence attenuation when conductors are PECs. When the frequency reaches the high value, the values of ε re approach each other, the values of ReZ drop sharply and the values of ImZ change obviously. The question arises as to how far g would have an effect on the propagation characteristics. In Fig., the results of ε re, α, ReZ and ImZ verus g from. mm to 9. mm are given. As can be seen from Fig., the values of ε re and α decrease with increased g. When g reaches about 4 mm or more, the curves keep stable. From Fig., we can see trends of ReZ and ImZ verus g are different, when calculated frequency is different. For instance, the curves of ReZ and ImZ are shown in Fig. at a low frequency 3 GHz and a high one 11 GHz. In general, those characteristic parameters all converge at certain values when the width of ground-planes is up to a high value. 2 Effective dielectric constant 6. re ε 4. 4 MOL HFSS Fig.9 in Reference d/λ Figure 2: Computed ε re as a function of the reciprocal of the free space wavelength with the conductor thickness as a parameter. L/d 4, h 1 /d 14., h 2 /d 14., w/d.4, w + 2s/d 2.4, w + 2s + 2g/d 1, ε r 9.6, f 3 GHz. ReZ ohm ReZ by HFSS ReZ by MoL ImZ by HFSS ImZ by MoL.8.6 ImZ ohm Frequency GHz Figure 3: Computed characteristic impedance Z against frequency by MOL compares with HFSS. L 2 mm, h mm, h mm, w.3 mm, s.14 mm, d.24 mm, ε r 9.9, tan δ.2, g 1 mm.

5 PIERS ONLINE, VOL. 6, NO. 1, 21 Effective dielectric constant ε re 6 g.6mm g1mm g9mm Frequency GHz 4 Re Z ohm Attenuation constant α Np/m g.6mm g1mm g9mm Frequency GHz.1 ImZ ohm. Figure 4: ε re, α, ReZ and ImZ against frequency from 1 GHz to GHz, L 2 mm, h mm, h mm, w.3 mm, s.14 mm, d.24 mm, ε r 9.9, tan δ.2, g.6 mm, 1 mm, 9 mm ε re ε re at 3GHz α at 3GHz. α Np/m.118 ReZ ohm 48 ReZ at 11 GHz ReZ at 3 GHz ImZ at 11 GHz ImZ at 3 GHz ImZ ohm g mm g mm Figure : ε re and α against g at the frequency 3 GHz, ReZ and ImZ against g at the frequency 3 GHz and 11 GHz, L 2 mm, h mm, h mm, w.3 mm, s.14 mm, d.24 mm, ε r 9.9, tan δ.2, g varies from. mm to 9. mm, and the step is. mm. 4. CONCLUSION In this paper, the method of lines is used for analyzing the propagation characteristics of CPW with finite ground-planes in a large range of frequency. The MoL with nonequidistant discretization provides accurate results with high computational efficiency. The process to calculate some propagation characteristics especially characteristic impedance of CPW is introduced and results with frequency from 1 GHz to GHz and finite ground-planes width from. mm to 9. mm are given. The effective dielectric constant, attenuation constant and characteristic impedance with respect to the frequency and finite ground-planes width are discussed and results agree well. The effects of finite ground planes have been investigated and the results may be useful for the design of the circuits. ACKNOWLEDGMENT This work was supported by basic research item of National Key Lab of Electronic Measurement Technology of China. REFERENCES 1. Browne, J., Broadband amps sport coplanar waveguide, Microwaves & RF, Vol. 26, No. 2, , Liang, G.-C., Y.-W. Liu, and K. K. Mei, Full-wave analysis of coplanar waveguide and slotline using the time-domain finite-difference method, IEEE Trans. on MTT, Vol. 37, No., , Pregla, R. and W. Pascher, Numerical Techniques for Microwave and Millimeter Wave Passive Structures, John Wiley & Sons, Inc, New York, Simons, R. N., Coplanar Waveguide Circuits, Components, and Systems, John Wiley & Sons, Inc, 21.