IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 5, MAY 2002 1245 An Improved Thin-Wire Model for FDTD Riku M. Mäkinen, Member, IEEE, Jaakko S. Juntunen, Member, IEEE, and Markku A. Kivikoski, Member, IEEE Abstract—An improved thin-wire model for the finite-difference time-domain method is proposed. The new model can be used to accurately model straight wire sections connected to other metal structures. In addition, the model includes the effect of charge ac- cumulation at wire end caps. The end-cap model is based on con- servation of charge and Coulomb’s law. Using the end-cap model, unconnected wires such as wire antennas are also accurately mod- eled. The results indicate a significant improvement in predicting the resonance frequency of a dipole antenna. Index Terms—FDTD, sub-cell model, thin-wire model. I. INTRODUCTION S UB-CELL models are used in finite-difference time-do- main (FDTD) method to describe small geometrical details without resorting to a large number of smaller cells. Sub-cell thin-wire models are widely used in antenna problems such as modeling of wire antennas or as a part of the antenna or waveguide feed structure. In either case, the computed input impedance or scattering parameters are affected by the accuracy of the thin-wire model. In addition to the wire radius, accurate modeling of wire length should be considered if the wire ends are not connected to other structures. In the standard thin-wire model [1] static field distribution is assumed for the scattered field components adjacent to the wire. The sub-cell geometry of a wire is modeled using the con- tour-path integral formulation of FDTD. Modifications to the standard model have been proposed, such as in [2], where fur- ther assumptions on the field distribution near the wire ends are made. A different approach is presented in [3], where FDTD is formulated as the time-domain finite-element method, and static field solutions are incorporated by modifying the basis functions near the wire. A filamentary hard source and a filamentary perfect elec- tric conductor (PEC) wire implemented by setting an electric field component on a wire axis identically to zero are known to have a finite effective radius . A method to obtain of a two-dimensional (2-D) filamentary hard source was pre- sented in [4]. By comparison to a frequency-domain Green’s function, the value of was found to be between and , where is the cell size. Due to the fact that a filamen- tary PEC wire is effectively a hard source with the excitation Manuscript received December 28, 2000. This work was supported in part by the Finnish Graduate School for Electronics, Telecommunications, and Au- tomation and in part by the Nokia Foundation. R. M. Mäkinen and M. A. Kivikoski are with the Institute of Electronics, Tampere University of Technology, FIN-33101 Tampere, Finland (e-mail: riku.makinen@tut.fi; markku.kivikoski@tut.fi). J. S. Juntunen was with the Radio Laboratory, Helsinki University of Tech- nology, FIN-02015 TKK, Finland. He is now with the Aplac Solutions Corpo- ration, FIN-00370 Helsinki, Finland (e-mail: jaakko.juntunen@aplac.com). Publisher Item Identifier S 0018-9480(02)04067-X. function identically set to zero, the procedure of [4] can be used to test the accuracy of a 2-D thin-wire model. This is done in [5], where it is shown that the standard thin-wire model [1] does not model the wire radius accurately in 2-D. In addition to accurate modeling of the wire radius, modeling of wire ends has a profound effect on the overall accuracy in modeling of wire antennas. Both the standard wire model and the three-dimensional (3-D) version of [5] have been shown to suffer from the coarseness error affecting the length of the wire [6]. A considerable improvement compared to the stan- dard thin-wire model was achieved by introducing further as- sumptions on the field distribution at the antenna ends based on NEC-computed data [2]. However, only the wire ends are con- sidered in [2]. In [5], a 2-D thin-wire model that accurately describes the wire radius was proposed. The 2-D model reduces to a filamen- tary PEC wire when the nominal wire radius is set to , which is in agreement with the result given in [4]. In this paper, the 2-D thin-wire model of [5] is extended into three dimensions. In addition, an algorithm to model the wire end caps is proposed. Instead of using precomputed field distributions for field components near wire ends, as in [2], we begin with sim- ilar assumptions that are used in NEC-4 to model the wire end caps [7]. The proposed model describing the effect of charge ac- cumulation at the wire ends is based on conservation of charge and Coulomb’s law. II. IMPROVED THIN-WIRE MODEL The proposed model is based on the contour-path integral for- mulation of FDTD, where the integral form of Ampere’s law and Faraday’s law are enforced locally in each Yee cell [1]. The Ampere’s law in free space is given by (1) where surface is a dual-grid cell facet bounded by contour formed by dual-grid cell edges. Similarly, the Faraday’s law in free space is given by (2) where surface is a primary-grid cell facet bounded by contour formed by primary-grid cell edges. The basic idea in the standard thin-wire model [1] is that it incorporates static field solution into FDTD. This is also the case with the proposed model. While the static field distribution is only used as an approximation, it is consistently enforced on all field components involved in the wire models [8]. 0018-9480/02$17.00 © 2002 IEEE