## 1. Introduction

[2] In this work, the problem of approximating the continuous-spectrum (CS) current excited by a finite source on a microstrip line is addressed, on the basis of an asymptotic evaluation of the relevant spectral-domain integral representation. As is well known, the CS current is responsible for spurious effects (i.e., attenuation and interference) in microwave planar circuits, whose operation is based on the excitation of a single bound mode [*Freire et al.*, 1999; *Mesa and Jackson*, 2002]; moreover, undesired crosstalk between neighboring circuits may also be produced [*Bernal et al.*, 2003]. On the other hand, accurate knowledge of the CS current is necessary in order to evaluate the radiated field in planar traveling-wave antennas, e.g., leaky-wave antennas [*Tamir and Oliner*, 1963; *Oliner*, 1987].

[3] In general, the total current excited on a planar structure can be represented as an inverse Fourier transform with respect to the spectral variable *k*_{z}, associated with the longitudinal *z* direction of the waveguide under analysis [*Di Nallo et al.*, 1998]. We refer here to the basic microstrip structure shown in Figure 1. The relevant spectral integrand has branch points and pole singularities [*Mesa et al.*, 1999] (see Figure 2); by deforming the integration path along the Sommerfeld branch cuts, the total current can be represented as the sum of the residue contributions of a finite number of poles, associated to the bound modes of the structure, and the contribution of the integral around the branch cuts, which constitutes the CS current [*Mesa et al.*, 2001].

[4] By means of a further contour deformation along the vertical steepest-descent paths (SDPs) through the branch points, the CS can be expressed as the sum of the residue contribution of the captured (i.e., physical) leaky-wave (LW) poles, if any, and the contribution of the integrals along the SDPs, which is known as the residual wave (RW) [*Mesa et al.*, 2001]. Such a representation is particularly useful in those cases when the contribution of a single LW pole allows us to accurately represent the CS. However, there are frequency regions in which the LW poles are not physical (spectral-gap regions [*Baccarelli et al.*, 2002, 2004]), and therefore the RW is the only contribution to the CS. In any case, since the LW has an exponential decay while the RW has an algebraic asymptotic decay with the longitudinal *z* coordinate [*Mesa et al.*, 2001], the latter is the dominant contribution to the CS at large distances from the source.

[5] In the case of a microstrip line, it has been shown in *Mesa et al.* [1999] that a logarithmic-type branch point is located in the *k*_{z} plane at the *k*_{0} free-space wavenumber, while square-root-type branch points are located at the wavenumbers of the TM and TE modes of the background structure; these TE and TM modes can be either above-cutoff modes, with a real propagation constant, or below-cutoff modes, which may be leaky waves with a complex propagation constant. Consequently, the RW current excited on a microstrip line can be expressed as the sum of a finite number of distinct components: a free-space RW current (FSRWC), associated to the *k*_{0} branch point; bound-mode RW currents (BMRWC), associated to the above-cutoff TM and TE modes of the background structure; finally, leaky-mode RW currents (LMRWC), associated to the physical leaky modes of the background structure, whose contribution decays exponentially with *z* and therefore can typically be neglected.

[6] The aim of the present work is to provide a rigorous asymptotic study of both the TM_{0}-BMRWC and the FSRWC for a microstrip line excited by a delta-gap source in a frequency region that extends through the entire spectral gap of the dominant leaky mode, in order to obtain an analytical representation of the CS current. In particular, by means of suitable integral evaluations in the complex plane of the transverse spectral variable *k*_{x} via Cauchy Integral Theorem, it will rigorously be shown that both TM_{0}-BMRWC and FSRWC have an algebraic asymptotic behavior for large values of the longitudinal *z* coordinate: the former as *z*^{−3/2} and the latter as *z*^{−2} [*Jackson et al.*, 2000b; *Mesa et al.*, 2001]. Moreover, as it will be made clear in the analysis, such different behaviors are related to the square-root or logarithmic nature of the involved branch points. An explicit expression for the numerical coefficient arising in the asymptotic expansion obtained through Watson's Lemma will be provided in a simple closed form for both the TM_{0}-BMRWC and the FSRWC. In order to obtain a good agreement also very close to the source, a more accurate formulation, which takes into account the presence of one pole singularity, will also be derived for the TM_{0}-BMRWC.

[7] The paper is organized as follows. Analytical derivations are reported in section 2 for both TM_{0}-BMRWC and FSRWC, on the basis of a simple spectral-domain representation of the microstrip current obtained through the method of moments [*Di Nallo et al.*, 1998]. A discussion on the proposed formulations is given in section 3. Numerical results for two different microstrip structures at frequencies inside the spectral gap of the dominant leaky-mode are presented in section 4. Some concluding remarks are provided in section 5.