1770 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES,VOL. 52, NO. 8, AUGUST 2004 Crosstalk Between Two Microstrip Lines Excited by a Gap Voltage Source Joaquín Bernal, Francisco Mesa, Member, IEEE, and David R. Jackson, Fellow, IEEE Abstract—The crosstalk between two microstrip lines is studied when one of the lines (the source line) is excited by a gap voltage source. In particular, the current induced on the passive line (the victim line) due to electromagnetic coupling to the source line is studied as a function of line separation and frequency for different permittivities. Results are also presented for the case of the source line excited by a vertical electric dipole to explore the effect of different source excitations on the crosstalk current. The current is calculated using a semianalytical method, which allows for an examination of the constituent current components on the lines (the bound-mode and continuous-spectrum currents) so that the physical mechanisms of coupling can be explored. The calculation is performed in an efficient manner using a mixed-potential inte- gral-equation formulation with complex images. Index Terms—Complex images, coupled lines, crosstalk, integral equations, leaky modes, microstrip, mixed potential, printed-circuit lines. I. INTRODUCTION T HE EXISTENCE of leaky modes on printed-circuit lines and the spurious effects that may be caused by them has been the subject of considerable interest [1]–[10]. Leaky modes are one part of the “continuous-spectrum” (CS) current [5], [11] that is excited by a practical source or discontinuity on a printed circuit line [11]–[17]. The CS current, which corresponds to ra- diation, generally increases with frequency and may become quite significant at high frequencies, causing significant spu- rious effects [13]–[17]. Recently, the nature of the current on an infinite microstrip line that is excited by a gap voltage source was studied [15]. It was shown there that, at low frequency, the source excites mainly a quasi-TEM bound mode (BM), as expected by simple transmission-line theory. A plot of the current amplitude versus distance along the line from the source is thus approximately constant at low frequency (neglecting the effects of conductor or dielectric loss). However, at high frequency, the current on the line exhibits spurious oscillations due to interference be- tween the BM and CS currents. The CS current consists of the Manuscript received July 23, 2003; revised December 14, 2004. The work of J. Bernal and F. Mesa was supported in part by the Spanish Ministry of Science and Technology and FEDER Project CICYT TIC2001-3163. The work of D. R. Jackson was supported in part by the Texas Advanced Research and Technology Program. J. Bernal is with the Departamento de Física Aplicada III, Escuela Superior de Ingenieros Industriales, 41092 Seville, Spain. F. Mesa is with the Grupo de Microondas, Departamento de Física Aplicada 1, Escuela Tecnica Superior de Ingeniería Informática, Universidad de Sevilla, 41012 Seville, Spain. D. R. Jackson is with the Department of Electrical and Computer Engineering, University of Houston, Houston, TX 77204-4005 USA. Digital Object Identifier 10.1109/TMTT.2004.831570 Fig. 1. Geometry of two coupled infinite microstrip lines with the source line (line 1) excited by a 1-V gap source, and the passive victim line (line 2) coupled electromagnetically to the source line (figure adapted from [18]). current of any physical leaky modes that exist, together with a “residual-wave current,” which is the leftover part of the CS current that is not representable in terms of the leaky modes. These interference effects may become very significant at high frequency. In this paper, the crosstalk between two infinite microstrip lines is examined, which significantly extends the work pre- sented in [18]. The first line (the source line) is excited by a 1-V gap voltage source at . The second line (the victim line) is passive, and is coupled electromagnetically to the first line. The geometry is shown in Fig. 1. The currents on both lines are calculated, with particular attention given to the crosstalk cur- rent on the victim line. The crosstalk current is studied as a function of the line separation and the frequency. Although the study is limited to a gap source or a vertical-dipole excita- tion of the source line, the general conclusions should be appli- cable to various geometries involving sources or discontinuities on interconnects or coupled lines and are, thus, expected to be applicable to a variety of structures that occur within high-fre- quency packages. A semianalytical spectral-domain technique is used to cal- culate the line currents so that a decomposition of the currents into the BM and CS parts is possible. This aids in the physical interpretation of the results, and allows for some important conclusions regarding the nature of the crosstalk current. The semianalytical method easily allows for the canonical problem of crosstalk between two infinite lines to be studied so that end effects can be ignored; this would be difficult to do using a purely numerical simulation in which the entire structure is discretized. The formulation is initially given in the spectral domain since this allows for a convenient extraction of the physics [19]. How- ever, the numerical calculation efficiency is improved by using a hybrid spectral/spatial formulation in which the mixed-poten- tial method together with discrete complex images is used to calculate the necessary reactions. The discrete complex-image technique (DCIT) is used to obtain a very good approximation for the kernel of the integral equation in the spatial domain. 0018-9480/04$20.00 © 2004 IEEE