## 1. Introduction

[2] An important consideration in the study of electromagnetic compatibility is determining the response of a transmission line that is illuminated by an incident electromagnetic (EM) field. Typically, this means determining the voltages and currents induced at the terminations of the line. It is necessary to understand this response in order to avoid performance degradation or the possibility of damage to equipment that is connected to these lines. At the same time, to avoid susceptibility, a line should not be over designed thereby adding unnecessary cost, weight, and size. Such lines may be found in computer or communication systems or as interconnects between systems within a vehicle such as an automobile or aircraft.

[3] The study of cable bundles can be performed by modeling them as multiconductor transmission lines (MTLs). The analysis of MTLs is well known and is a basic tool for the analysis of such lines [*Paul*, 1994]. The analysis begins by determining the per-unit-length (PUL) parameters of inductance, capacitance, resistance, and conductance that are necessary for determining solutions of the line. In the work of *Paul* [1994], the MTL equations are solved by use of the chain parameter matrix, which is a representation that relates the voltages and currents on the line. The chain parameters are also used in a simple method for solving a nonuniform MTL (NMTL.) In that method, a NMTL is approximated as a series of smaller uniform lines [*Paul*, 1994; *Omid et al.*, 1997; *Mao and Li*, 1992]. Each of these small uniform lines is characterized by a chain parameter matrix. From these, an overall chain parameter matrix for the line can be found. Because the NMTL equations can only be solved analytically for a few cases, this method is a useful technique and is used in this work.

[4] The effects due to illumination by an external EM field can be included in the MTL equations by the incorporation of additional source terms. The solution can again be derived in terms of the chain parameter matrix resulting in a form similar to that of nonillumination. There are three basic models for including external illumination [*Nucci and Rachidi*, 1995]. Each of the three models adds additional source terms to the MTL equations due to excitation by the external field. Though the models differ in their interpretation of the source terms, they can be shown to be equivalent. In the first model, referred to as the Taylor model, the source terms are due to both incident electric and magnetic fields that give rise to distributed voltage and current sources on the line [*Taylor et al.*, 1965]. In the second model, referred to as the Agrawal model, the source terms are due to electric fields only that give rise to distributed voltage sources as well as lumped voltage sources at the terminations [*Agrawal et al.*, 1980]. In the third model, referred to as the Rachidi model, the source terms are due to magnetic fields only that give rise to distributed current sources as well as lumped current sources at the terminations [*Rachidi*, 1993].

[5] In practical applications, it may be necessary to model a MTL as a random cable. This would reflect realistic situations in which the exact position of a cable or the exact positions of wires within a cable are unknown or can vary. Some of the work addressing random transmission lines includes the following. In the early work of *Morgan and Tesche* [1978], the problem of a random cable is confronted and a method of statistical analysis is discussed. In the work of *Capraro and Paul* [1981], experimental data were obtained for a random cable by performing crosstalk measurements on a cable bundle that was successively wrapped and unwrapped to produce variations. In the work of *Ciccolella and Canavero* [1995], the experiment of *Capraro and Paul* [1981] was modeled using a statistical simulation. In that work random cable bundles were generated and the corresponding NMTL equations were solved numerically. It was found that the results of the simulation agreed well with the experimental data. In the work of *Shiran et al.* [1993], a probabilistic model was developed on the basis of a simplified deterministic model. In that work, the parameters of the simplified MTL solution were modeled as random variables, and the probability density function (PDF) of the crosstalk in the bundle was determined. In the work of *Salio et al.* [1999], a method for generating random cables was proposed. The resulting NMTLs were modeled as a cascade of uniform lines, which were solved, and statistical results for the crosstalk found. In the work of *Pignari and Bellan* [2000] and *Bellan and Pignari* [2001], the effects of a random field were considered. In that work, a two-conductor transmission line and a MTL, both of which were uniform and lossless, were subject to a randomly oriented plane wave. Numerical and analytical solutions were developed, with the analytical solutions being for the electrically short case.

[6] In this work, cable bundles are modeled as MTLs that are excited by an external EM field. The random nature of the cable bundle is modeled by segmenting the cable into a number of small uniform sections that approximate the overall cable. The orientation of these sections is varied randomly to generate an overall random cable. The source term generated at each section due to the external field depends on the orientation of that section with respect to the field.

[7] By generating a sufficiently large number of random cables, the statistical nature of the response to an external field can be determined. The statistical response is shown by plotting the cumulative distribution function (CDF), and the PDF of the frequency response of the cable. This paper is organized as follows. In section 2, MTL theory is briefly reviewed. In section 3, a model is first developed for the case of a uniform MTL and then for a NMTL. In section 4, the model of section 3 is applied to a ribbon cable with the results shown in section 5.