ON THE USE OF THE SURFACE IMPEDANCE APPROACH IN THE QUASI-TEM ANALYSIS OF LOSSY AND SUPERCONDUCTING STRIP LINES R. Marques, zyxwvutsrqpo J. Aguilera, F. Medina, and M. Horno zyxwvutsr Universidad de Sevilla Departmento de Electronica y Electromagnetismo Avd Reina Mercedes sin 41012-Sevilla KEY TERMS zyxwvutsrqponm Lossy microstrip, superconductor, MMIC ABSTRACT In this arricle we propose a mixed quasi-TEM surface-impedance zyxwvutsr (SI) approach for the analysis of metallic losses in MMICs and of superconductivity effects in microstrip lines. Our results show that the quasi-TEM-SI technique is a very interesting choice to compute these effects. zyxwvutsrqp 0 1993 John Wiley & zyxwvutsrqp Sony, Inc. 1. INTRODUCTION zyxwvutsrqpo The applicability of the quasi-TEM model to MMIC lines involving lossy substrates has been recently investigated by some authors in [l]. It has been shown in [l]. that carefully computed quasi-TEM data are accurate enough for a wide range of frequencies, materials, and dimensions. However, conductor losses could modify this conclusion since they are responsible for the existence of a nonzero z-directed electric field. This field is more important in MMICs than in MIC structures, because the strip thickness is often of the same order as the skin depth penetration in MMIC lines [2]. This later fact precludes the application of the well-known Wheel- er's inductance rule, but it does not preclude the application of other quasi-TEM approaches. The suitability of the quasi- TEM analysis in MMICs is discussed in [2]. On the other hand, the surface impedance (or complex resistive boundary conditions) concept, first applied to the analysis of superconducting transmission lines [3], has been fruitfully applied 'to the analysis of conductor losses in planar lines in the full-wave framework [4-61. However, to our knowledge, no quasi-TEM formulation has been reported in- corporating this useful concept. In this article we use both the quasi-TEM formulation and the surface impedance model to evaluate microstrip conductor losses, developing a mixed quasi-TEM-SI approach. In order to investigate the feasibility of this approach, we have also implemented a full-wave anal- ysis using the same surface impedance model. Comparison of the results provided by the two techniques shows that the quasi-TEM-SI technique is a very interesting choice to com- pute conductor losses in MMICs, since it is much simpler and less time consuming than the full-wave formulation. In ad- dition, the same approach is also applied to the study of superconducting microstrip lines. 2. THE QUASI-TEM-SI APPROACH Let us consider the microstrip line in Figure l(a). Following [2] we will consider H, negligible in the whole structure, E, < /E,/ outside the conductors, and E, negligible inside the conductors. So we can neglect the external E, to evaluate E,, taking in the first-order approach V x zyxwvuts E, = 0, V x H, = 0 outside the conductors (quasi-TEM approach). Therefore these fields are derived in the Lorentz gauge from an electrical potential +(x, y)e-jP, and a z-directed magnetic potential A,(x, y)e-IYZ (y being the complex propagation constant along Figure 1 model (a) Geometry of the problem. (b) Surface impedance the line). Outside the conductors both 4(x, y) and A,(x, y) must satisfy the Laplace equation for isotropic media, or its extension [7] for anisotropic media. On the other hand, 4 is taken to be constant inside the conductors (4 = V in the strip, 4 = 0 in the ground planes), since the transverse electric field has been neglected in these regions. The longitudinal electric field can be written, in the first- order approach, in terms of the aforementioned + and A; potentials: Now substituting (1) in Maxwell equations, we obtain the equation governing the distribution of A, inside the conduc- tors: VfA, - jwpaA, = -jpayV, inside the conductors (a and p are the conductivity and permeability of the con- ductor; w is the angular frequency). At this point it can be recognized that the electromagnetic problem has been decoupled into an electrostatic problem and a magnetic problem. The latter remains magnetostatic outside the conductors (V:A, = 0) and reduces to the diffusion equation (2) inside the conductors. The conductor losses are related to the magnetic part of the problem. The rigorous computation of the line resistance R and inductance L implies the solution of Eqs. (2) inside the conductors. However, we choose here an alternative approximate method making use of the surface impedance concept. First we substitute the nonzero thickness strip by a planar strip carrying an equivalent surface current zyxwv Js(x). A similar equivalent surface current, JFK is defined for the ground plane(s). Therefore, we have re- duced the original problem to the equivalent planar problem in Figure l(b). The equivalent surface currents J,y and Jsx are linearly related to the value of E, on the bottom of the strip and on the ground surface, respectively. This linear relation- ship is explicitly expressed in terms of a proper surface imped- ance boundary condition: MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 6, No. 7, June 5 1993 391