Matlab Simulation of Transmission Lines with Skin Effect via Fractional Telegraph Equations and NILT Lubomír Brančík, Aslihan Kartci, Nawfal Al-Zubaidi R-Smith Department of Radio Electronics Brno University of Technology Brno, Czech Republic brancik@feec.vutbr.cz, kartci@feec.vutbr.cz, alzubaidi@phd.feec.vutbr.cz Abstract—The paper deals with the methods of simulation of signal propagation on transmission lines when skin effect is taken into account. Such simulations are useful when solving signal integrity issues in todays high-speed electronic circuits where interconnects play an important role. On conditions of continuing increase of the frequency of signals to be transferred, the real interconnects are modelled as transmission lines. Then, in case of sufficiently high frequency band, additional losses caused mostly by a skin effect exist. Two basic approaches can be considered for the simulation, based on either the Laplace-domain or time- domain formulations of fractional telegraph equations, and both accompained by proper numerical techniques. The methods are intended to be utilized for both research and education, while respective Matlab language programs can be utilized for PC exercises. The educational aspects of this research are important, as signal integrity issues and related methods fit into a doctoral course Modern Electronic Circuits Design at our university. Keywords—Matlab simulation; transmission line; skin effect; fractional telegraph equation; Laplace transform I. INTRODUCTION The paper is devoted to problems of simulation of signal propagation on transmission lines (TL) that play an important role in the high-speed electronic circuits where interconnects between functional blocks, e.g. on the printed circuits boards (PCB) or even inside the chips, are present. As the frequency of signals is continuously increasing the wave effects cannot any more be neglected and interconnects are processed as systems with distributed parameters while the transmission line theory is utilized [1]. Besides, considering the real lossy transmission lines the increasing frequency leads to additional losses due to the frequency dependencies of the TL per-unit-length (p.u.l.) parameters. Most important is a skin effect affecting the series resistance and inductance [2], [3]. The paper discusses two approaches for the simulation of lossy transmission lines influenced just by the skin effect. The first method is based on usual approach when the skin effect is described in the s-domain solution through K√ term added to the TL series impedance [3]. It is enabled as we consider only linear transmission lines in our study that is why a Laplace transformation can be used to solve given telegraph equations. Here, after incorporating boundary conditions a proper method for numerical inversion of Laplace transforms (NILT) can be applied to get the time-domain solution [4]-[7]. The second method is based on the theory of fractional differential equations (FDE) which started to play an important role in last decades to more accurately model a number of real physical processes [8]-[11]. Among them also the skin effect can be considered. When formulating the time-domain description back from its s-domain representation, the above mentioned √ term will lead to a half-order derivative, ∂ 1/2 /∂t 1/2 , resulting in respective fractional telegraph equation, see e.g. [12], that is a partial differential equation with a fractional-order derivative by time. Finite-difference time domain (FDTD) techniques are being adapted for the solution, taking into account the rules for fractional differentiations and their approximations [8]. Beside, special FDE numerical methods freely available on the Matlab Central File Exchange are tested [13]. Both approaches under investigation are continuously being programmed in Matlab language. For this purpose a problem-oriented graphical user interface (GUI) was created for easily inserting all parameters. The topic belongs to a broader area of the signal integrity issues [1], that is part of the doctoral course Modern Electronic Circuit Design [14], at the Brno University of Technology, the Faculty of Electrical Engineering and Communication. When explaining relevant methods the Matlab language environment is widely used for demonstration. Besides, the students of all levels working on their final projects often utilize the same software. Thus, these Matlab functions will be useful not only in the area of the research but also in the educational process. II. DESCRIPTION OF TRANSMISION LINE SYSTEM We consider a transmission line system with a nonuniform TL of a length l, terminated by linear circuits, see Fig. 1. Fig. 1. Linear system with nonuniform transmission line. This research was financially supported by the Czech Science Foundation under grant 15-18288S, and by Czech Ministry of Education in frame of National Sustainability Program under grant LO1401. For research requirements, infrastructure of the SIX Center was used.