An Efficient Algorithm for Planar Circuits Design - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 8, AUGUST 2010

3441

An Efficient Algorithm for Planar Circuits Design A. Serres1 , G. Fontgalland1 , J. E. P. de Farias2 , and H. Baudrand3 , Fellow, IEEE Laboratório de Eletromagnetismo e Microondas Aplicados-L.E.M.A., UFCG, Campina Grande, PB, CEP. 58.429-140, Brazil Instituto de Estudo Avançados em Comunicações-I.E.C.O.M., UFCG, Campina Grande, PB, CEP. 58.429-140, Brazil L.A.P.L.A.C.E.-G.R.E., Toulouse Cedex 7, 7122-31071, France An efficient algorithm for planar circuits design based on the wave concept (WCIP) is formulated and applied. The algorithm is based in the use of a recursive relationship between incident waves and reflected waves from a discontinuity plane instead of an integral equation method. The advantage here are that no matrix inversion is required, the convergence is insured independently of the kind or numbers of interfaces of the structure and well adapted to multi-layered circuit characterization, antenna coupling over concentric cylinders. The method herein presented solves for the current density and the tangential electrical field on the interface. The robustness of this formulation permits to solve, independently of the frequency or physical size, several problems by applying dual quantities. In order to simplify calculations and accelerate the convergence, a two-dimensional fast modal transformation (FMT) algorithm using periodic walls is used with the 2D-FFT algorithm. As example, the simulation results for two different structures, an open microstrip line and a microstrip patch antenna, are presented. Index Terms—Dual quantities, fast modal transform, iterative procedure, WCIP.

I. INTRODUCTION

T

HE wave concept is a well-established procedure to treat several electromagnetic problems [1], it is used in the Transverse Line Matrix (T. L. M.) method [2]. From the transmissions lines (T. L.) theory where the dual quantities are the current and the voltage, it is possible, by analogy, to use the current density (current) and the electric field (voltage). Here, the advantage is to have fields that are collinear in the interface plane. In this paper, the Wave Concept Iterative Procedure (WCIP) is applied to solve problems as into the waveguides structures, taking in account the Transverse Electric (TE) and Magnetic (TM) modes. A tendency to full wave simulations for microwave systems instead of analyzing components independently is growing. For example, several analytical and numerical techniques are reported in the literature for the analysis of monolithic microwave integrated circuits (MMICs), among them: the method of moments [3], the finite elements method and the finite difference time domain method [4]. For the reduction of development times and costs of MMICs, it is of primary importance the use of fast and efficient software tools, which can accurately predict the electrical behaviour of a device. The computational time of these numerical techniques is the limiting factor in the practical design of microwaves circuits. This method is not conditioned by the complexity of the circuit design and was proved to be particularly interesting for planar multi-layers circuits [5], [6]. The WCIP approach consists in separating the structure under study into interfaces with upper and lower homogeneous media. The boundary conditions on the interface are represented by the diffraction operator, S, and in the homogeneous media by the Manuscript received December 23, 2009; revised February 12, 2010; accepted February 13, 2010. Current version published July 21, 2010. Corresponding author: A. Serres (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2044150

reflection operator, . They are defined in spatial and modal domains, respectively. The two conditions for existence of the Wave Iterative formulation are: first, the partition of the global domain in two subdomains: • interior or spatial domain (interfaces or lumped elements); • exterior or spectral domain (medium or propagation). Second, for the physical behavior, it is necessary to have dual quantities: • current (I)—voltage (V): like in T. L.; • electric field (E)—magnetic field (H); • current density (J)—electric field (E); —electric field (E); • surface current —magnetic field (H); • magnetic current —electric potential (V); • charge density • or a function and its derivative. For example, with dual quantities and , the definition of waves A and B, is (1-2) with a scalar, often homogeneous with an impedance. In the work reported herein, the analysis and simulation are made in the same way as the electromagnetic field propagation into a rectangular waveguide. The two considered example cases treated by the WCIP formulation are an open microstrip line and a microstrip patch antenna with periodic walls. Section II provides the theoretical formulation used in the development of the WCIP. Simulation results are presented in Section III. Finally, conclusions are included in Section IV.

II. FORMULATION OF THE WCIP The Wave Concept method described here is based on full wave transverse formulation, where the dual quantities current density and electric field are considered.

0018-9464/$26.00 © 2010 IEEE

3442


is the current density on the metal in each medium where with the iteration . is the electric field in each medium where the excitation is carried out. The value considered is taken and the width and only on the domain source’s (ds) with the length of the source. Applying the periodic boundaries conditions to the problem, the FMT of the transverse components of the spatial incident , is given by wave,

(7) Fig. 1. Iterative Procedure illustration for a planar circuit.

and its The incident (A) and reflected (B) waves are calculated from the tangential electric (E) and magnetic (H) fields, on the interface

(3-4)

(8)

where indicates the medium 1 or 2 corresponding to a given is the characteristic impedance of the same interface . medium and is the surface current density vector given as

and are the 2D-FFT and 2Dalgowhere rithms operators, respectively, and and are the waveguide dimensions. The usually used WCIP scheme is very simple. Two operators relating incident and reflected waves in the spatial domain and in the spectral domain governs the iterative procedure. It can be represented through two equations

(5) where is the outward vector normal to the interface. Let us consider a single, but general, interface problem. In this case, a thin metallic plate is at the interface between two medium, enclosed by a rectangular waveguide with transversal cut shown on Fig. 1. The operator is described in the modal domain and assigns the propagation boundary conditions at upper and lower interface. The S operator expressed in the spatial domain assigns boundary conditions at the interface plan and represents the different possible subdomains (dielectric, metal and source). Then, the air-dielectric interface plan is divided into cells (named pixels), forming an uniform grid, used to disgenerates two waves, cretize each subdomain. The source one in each side of the interface, indicated by the arrows in Fig. 1. The spatial and modal waves are directly deduced from each other with the help of Fast Modal Transformation (FMT) and . The decomposition of the elecits inverse transform tromagnetic wave in guided modes propagating in waveguide with periodic wall (TE and TM modes) takes place by the use of this Fast Modal Transformation. The FMT is comprised of the 2D-FFT algorithm and transformation from the spectral to the modal domain. A Fast Modal Transform (FMT) and its inverse ensure conversions between the two domains. The procedure is repeated until convergence of the input admittance of the strucis obtained. This parameter of convergence is directly ture deduced of the dual quantities of the problem to solve (6)

(9) (10) is the local source of the circuit. where Finally, it is possible to express the boundary conditions in terms of waves on each cell and the tangential electric and magnetic fields can then be calculated from (11) (12) The conditions for existence of a Wave Iterative formulation in this case are the decomposition of the global domain in two subdomains: the spatial domain, with the diffraction operator S and the modal domain, with the reflection operator . The other condition is to have dual quantities for the physical behavior that are the current density (J) and electric field (E). III. SIMULATIONS RESULTS In order to evaluate the performance of the proposed tool, two case studies were performed. The cases considered permit to treat problems of coupling (open microstrip) and propagation (patch antenna). The simulation results for the two different structures are given in Sections III-A and III-B.

SERRES et al.: AN EFFICIENT ALGORITHM FOR PLANAR CIRCUITS DESIGN

3443

Fig. 2. Open microstrip line. Fig. 4. Microstrip patch antenna.

Fig. 3. Input impedance in Ohms for an open microstrip line as a function of frequency in Hertz.

Fig. 5. Input reflection coefficient jS11j.

A. Open Microstrip Line

B. Microstrip Patch Antenna

An open microstrip line in a waveguide of dimension 32 mm 8 mm with periodic walls was simulated with the Matlab software. The microstrip line lies on a substrate with and height of 1 mm. The simulated source has dimensions of 2 mm 2 mm, while the simulated microstrip line is 2 mm wide, and 25 mm length. , is shown in Fig. 3. The The computed input impedance, imaginary part is in conformity with the theory [7] where the analytical formulation for an open microstrip line is given by . As expected, the real part reaches a positive peak value at the line resonance frequency. A length of line resounds for the lengths corresponding to half-wavelength. The two first GHz and theoretical resonance frequencies are GHz. The two first computed resonance frequencies are GHz and GHz. The discrepancy when compared to the theoretical values, is caused by the limited number of pixels used and by a slower convergence close to the resonance. In this case 64 pixels 64 pixels and 400 iterations were used. All of the frequencies of resonance are multiple of the first frequency. All resonance frequencies are integer multiple of .

To analyze the propagation phenomenon of the electromagnetic field by the WCIP method a patch antenna with resonance frequency of 1.32 GHz is now considered. Its physical construction guarantees a characteristic impedance of 50 Ohms. The dimensions of the waveguide are 152 mm 152 mm, mm, mm and mm. The simulated source has dimensions of 4.75 mm 4.75 mm. The antenna lies on a and height of 0.5 mm. substrate with As can be seen in Fig. 5 (blue curve), the resonance frequency simulated result obtained by WCIP is very close to the measured frequency (at 1.317 GHz) [8]. This is in good agreement with the measured frequency presented in [8]. Therefore, it can be used to confirm that the method proposed herein gives rather satisfactory results, as they are closer to the measured ones than those of the simulations in [8], with a comparatively small computational effort. In this case 100 pixels 100 pixels and 1500 iterations were used. and the current density are presented The electric field in Figs. 6 and 7, respectively, at the first resonance frequency. The expected physical behavior can easily be observed. The slots of adaptation induce a concentration of current density

3444


IV. CONCLUSION

Fig. 6.

j

Jxj (A/m), f = 1:317 GHz.

In this paper, an implementation of the WCIP using periodic walls is presented. An open microstrip line and a microstrip antenna patch have been analyzed and simulated. From the formulation presented here it is easy to develop any other implementation program to treat different electromagnetic problem. The obtained results show that the method herein formulated and implemented in software, is suitable to electromagnetic analysis of planar circuits. Very good agreement among simulated results and physical implementation has been achieved. This formulation can be extended for several physical problems if the two conditions for existence of a Wave Iterative formulation are respected. ACKNOWLEDGMENT This work was supported by the National Program of Academic Cooperation—CAPES and CNPq, and COPELE/CEEI/ UFCG post-graduate program. REFERENCES

Fig. 7.

j

Exj (V/m), f = 1:317 GHz.

close to the antenna input. Due to the symmetry of the antenna, the electric field and the current density are uniformly distributed. At the frequency of resonance, the electric field Ex is concentrated on the edges of the source and on the two edges along the vertical axis, as illustrated in Fig. 7. The distribution of current density is in conformity with the theory. The half-wavelength for the first resonance frequency is on the edge of the patch, along the horizontal axis, as shown in Fig. 6.

[1] K. Kurokawa, “Power waves and the scattering matrix,” IEEE Trans. Microw. Theory Tech., vol. 13, pp. 194–202, Mar. 1965. [2] N. Fichtner, S. Wane, D. Bajon, and P. Russer, “Interfacing the TLM and the TWF method using a diakoptics approach,” in IEEE MTT-S Int. Microwave Symp. Dig., Jun. 2008, pp. 57–60. [3] M. M. Ilic, M. Djordjevic, M. Ilic, and B. M. Notaros, “Higher order hybrid FEM-MoM technique for analysis of antennas and scatterers,” IEEE Trans. Antennas Propag., vol. 57, pp. 1452–1460, May 2009. [4] R. Wang and J.-M. Jin, “A symmetric electromagnetic-circuit simulator based on the extended time-domain finite element,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2875–2884, Dec. 2008. [5] S. Wane and H. Baudrand, “A new full-wave hybrid differential integral for the investigation of multilayer structures including nonuniformly doped diffusions,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 200–213, Jan. 2005. [6] H. Zairi, A. Gharsallah, A. Gharbi, and H. Baudrand, “Analysis of planar circuits using a multigrid iterative method,” IEE Proc. Antennas Propag., vol. 153, no. 3, pp. 231–236, Jun. 2006. [7] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: Wiley/IEEE Press, Jan. 2001. [8] F. Surre, L. Cohen, H. Baudrand, and J. C. Peuch, “New approaches for multi-scale circuit analysis,” in Proc. 31st Eur. Microwave Conf., Sep. 24–26, 2001, pp. 1–4.