Analysis of Radiating Microstrip Structures Using the Contour Integral Method Utkarsh R. Patel*, Piero Triverio, and Sean V. Hum Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto Toronto, Ontario, Canada utkarsh.patel@mail.utoronto.ca Abstract—In this paper, we discuss the possibility of numeri- cally solving electromagnetic radiation problems from microstrip structures with only contour discretization of the radiating elements. We demonstrate that such a numerical method can be realized by exploiting the underlying physics of microstrip surfaces, and combining the so-called contour integral method with the equivalence principle. Such a numerical technique requires fewer unknowns and can potentially lead to significant computational savings in the simulation of large microstrip struc- tures such as arrays, reflectarrays, metasurfaces, etc. Preliminary results show that the technique can accurately predict the input impedance and radiation pattern of a patch antenna in an array environment. I. I NTRODUCTION The electromagnetic equivalence principle allows one to represent a complex field distribution inside a closed volume with equivalent electric and magnetic current sources on the enclosing surface. The equivalence principle motivated the development of the integral equation method (IEM) [1] which requires discretization of only the surface of a metallic scatterer. In comparison to finite element method (FEM) [2], which needs volumetric 3D discretization with number of unknowns (N ) of order O(n 3 ), the IEM needs only a 2D mesh which contributes to superior speedups and N ∼ O(n 2 ) † . However, for problems involving electrically large microstrip structures, such as reflectarrays, transmitarrays, and metasur- faces, the computational time and memory requirements can become unreasonable even with IEM. This motivates the need for reduced-order methods that facilitate further dimension- ality reduction. Such a method would significantly lower the number of unknowns and enable the simulation of large planar microstrip arrays, including metasurfaces. In the literature, contour-based methods have been success- fully applied to analyze guided-wave microwave structures [3] where radiation loss is of little interest. A contour-based method has been proposed for simulating patch antennas [4], but it requires prior knowledge of field distribution on the edges of the patch, and can therefore be applied only to resonant antennas where the cavity model is accurate [5]. In this paper, we discuss how the physical understanding of microstrip antennas can be combined with the equiva- lence principle [5] and the so-called contour integral equation † n is the number of basis functions per dimension J ≈ 0 z x y J =0 γ γ J, M CIM γ  r  r ′ O ˆ t ˆ n Ez ,Ht Fig. 1. Left panel: Assumed current distributions. Right panel: Equivalent contour and notation. #3 #2 feed #1 #4 Ground plane 0.5 3.14 11.86 9.06 2.94 2 Fig. 2. Dimensions (in mm) of the patch antenna array considered in Sec. IV. A coaxial feed is used to uniformly excite all elements. method (CIM) [3] to analyze radiating microstrip structures. The presented technique is general and does not assume any prior knowledge of field distribution on the radiating elements. II. ASSUMPTIONS Consider the geometry of a sample patch antenna shown in the left panel of Fig. 1. The following two assumptions hold when electrical spacing between the two metallic plates is small [5]: • The z-directed electric field is dominant compared to the z-directed magnetic field, which may be ignored. In other words, the fields satisfy the transverse magnetic field assumption. • If the two parallel plates are enclosed by an equivalent surface, then the main radiation mechanism is due to the equivalent currents on the periphery. The equivalent currents on the top and bottom surfaces may be ignored. Additionally, for thin dielectrics we may assume that the fields are invariant along z-direction, i.e. ∂E/∂z =0.