This work considers the problem of characterizing the response of a cable bundle in the presence of an external electromagnetic field when the orientation of the cable is not well defined or is unknown. To do this, cable bundles are modeled as multiconductor transmission lines. Because of the random nature of the cable, an exact solution is not in general possible. Therefore a solution is approximated by segmenting the cable into a number of small uniform sections that can be solved and combined to form an overall solution. The orientation of these sections is allowed to vary randomly, thereby modeling the various twists and bends that may be found in a practical application. The line is excited by an external electromagnetic field. The voltages and currents generated at the terminations of the cable are calculated using a numerical approach to solve the multiconductor transmission line equations in the frequency domain. Because the exact positioning of the cable is not known, statistical data for the response are needed. To obtain statistical data, a large number of randomly generated cables are generated and solved.
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 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.
 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 , 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.
 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].
 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 , the problem of a random cable is confronted and a method of statistical analysis is discussed. In the work of Capraro and Paul , 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 , the experiment of Capraro and Paul  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. , 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. , 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  and Bellan and Pignari , 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.
 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.
 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.
2. MTL Theory
 MTL theory is well established and is only briefly reviewed here. The equations that describe the voltages and currents on a MTL in the frequency domain are given by
where (z) and (z) are vectors describing the voltages and currents along the line. The PUL parameters are included in the impedance matrix and the admittance matrix . Source terms due to external fields are given by the vectors F(z) and F(z). For a MTL with n conductors plus a reference, the PUL matrices are of size n × n, and the voltage and current vectors are of size n. Solutions for these equations are given in the work of Paul  and can be expressed by use of the chain parameters which have the following form
where the chain parameters are formed by the n × n submatrices ij. The vectors (L) and (L) describe the voltages and currents at the load end of the line and the vectors (0) and (0) at the source end. The vectors FT(L) and FT(L) are derived from the external field source vectors. FT(L) and FT(L) can be written in terms of incident electric fields only, which was the form used in this work, and are given as follows
where 1n×n is the ones matrix of size n × n, the subscript r denotes electric fields parallel to the length of the line, the subscript t denotes fields tangent (in the direction of the reference wire to the nth wire) to the line, and dn is the distance from the reference wire to the nth wire in the tangential direction. The postscript i denotes incident fields. Though not shown here, there are also reflected and transmitted fields when a ground plane is present. For the case of plane wave illumination, which is used in this work, solutions for (4) and (5) can be found and are given as follows
where 1n is a ones vector of size n, E0 is the amplitude of the incident electric field, er and et are the parallel and tangential components, βr is the propagation constant of the incident electric field parallel to the wires, βt is a diagonal matrix containing the propagation constants of the incident electric field tangential to each of the n wires, d is diagonal matrix containing the distance between each wire and the reference wire, is a diagonal matrix containing the propagation constants on the wires, and is a matrix containing the associated eigenvectors. Details on these terms and their derivation can be found in the work of Paul  and Paul . The above expressions are for isolated wires. Though not shown here, similar expressions can be found for wires over a ground plane.
 The load and source terminations, which provide boundary conditions at the ends of the MTL, can be included by considering them as either Norton or Thèvenin equivalent circuits. The latter was used in this work and is given as follows
where S and L are vectors of size n accounting for voltage and current sources at the source and load ends of the lines. S and L are n × n matrices describing the impedance at the source and load. In the above, the term source simply refers to one end of the line while the term load simply refers to the other end of the line. It is not implied that an actual source exciting the line is present at the source end, and in general such a source may or may not be present at either end.
3. Model Development
 In the previous section the solutions (6) and (7) for the total source terms due to an external field were shown for the case of uniform plane wave illumination. In this section a model that solves for the case of an arbitrary field is developed. This is accomplished by dividing the transmission line into discrete sections and approximating the external field as a uniform plane wave at each section. In this way, an approximate solution can be obtained, with the approximation becoming more accurate as more sections are used. This is first performed for a uniform MTL, which is when the PUL parameters are constant, and then for the case of a NMTL.
3.1. Uniform MTLs
 The development of this model begins with the expression for the total source terms given in (4) and (5). It is assumed that the left most integral on the right hand side of these equations cannot be solved analytically. Therefore it is approximated by breaking the integral into small discrete sections. The external fields at each section are then approximated by a uniform plane wave, and therefore the source term for that section can be solved analytically as shown in (6) and (7). The overall solution is then the summation of the solutions for each discrete section. This is shown in Figure 1, which shows a transmission line of length L, divided into N sections. With reference to Figure 1 and (4) and (5) the source terms due to the electric fields along the wires can be approximated as
where the fields in the summation are uniform plane waves denoted by iupw. The integrals can be solved analytically giving
where Ei0 is the amplitude of the electric field for the ith segment, eir is the electric field component longitude to the line, βir is the propagation constant longitude to the line, and βit is the propagation constant tangent to the line. The source terms due to the tangential fields at the terminations have not been shown because they do not change from (4) and (5).
 Next, the case of a NMTL is considered. Like the uniform MTL, the NMTL is divided into discrete sections and at each section the external field is approximated as a uniform plane wave. In addition for NMTLs, the PUL parameters vary along the length of the line. However, an analytical solution only exists for a few cases of NMTLs. Therefore at each discrete section the NMTL is approximated as a uniform line and the PUL parameter for each section is constant. The MTL equations can then be solved at each section and an overall result found by linking the individual results together via the chain parameters. In this way, an approximate solution can be obtained, and again the approximation becomes more accurate as more sections are used.
 This is accomplished as follows. Consider a MTL that is divided into N discrete sections as shown in Figure 1. The currents and voltages at the source and load ends of the kth section are related via the chain parameters for that particular section as
 Because the output of the kth section is the input for the kth+1 section, the overall chain parameter matrix can be obtained by substituting the solution for each section into the adjacent section. The overall solution is given as
 This can be written more compactly as
 The source terms at each section, FT(zk) and FT(zk) can be found using (4) and (5). It has been assumed that the nonuniformity in the line can be approximated by dividing the line into N divisions. If (4) and (5) are used for the source terms, then it is further assumed that the nonuniform external field can be accurately modeled by the same N divisions. If this is not the case and finer divisions are needed for the external field, then (12) and (13) can be used instead.
4. Application to a Ribbon Cable
 In this section the previously described model is applied to the case of a random ribbon cable illuminated by an external field. The cable is random in that it has various undetermined twists and bends along its length. The twists and bends are modeled by dividing the cable into smaller uniform sections that approximate the overall cable. The orientation of each section with respect to the incident field is calculated. From this, the source term for each section is then determined. From these source terms, the overall solution is found. A large number of random cable realizations are generated and solved in order to obtain statistical data for the response. These data consist of the CDF and PDF of the frequency response of the cable.
 The cable considered here was a flat ribbon cable consisting of 28 AWG (7 × 0.127) stranded wires with a spacing of 1.27 mm, and insulated with polyvinyl chloride (PVC) with a dielectric constant of ɛr = 3.5. Any cable realization would have been suitable for this example, however a ribbon cable was chosen because it is widely available, commonly used, and has been studied in other work [Paul, 1994]. The PUL parameters for the cable were calculated using method of moments as described in the work of Paul .
 For this particular example using a ribbon cable, it is assumed that the wire spacing does not vary along the length of the line. Therefore the PUL parameters are constant and the cable is modeled as a uniform MTL despite the twists and bends. It is not possible for a real ribbon cable to bend without introducing some nonuniformity. Because the cable here is considered uniform, it allows for bends with small discontinuities as opposed to nonuniformity. To make this approximate model accurate, several factors were taken into account. First, the degree of the bends was limited to avoid large discontinuities. Second, it was recognized for a ribbon cable, that random orientations along the cable are not entirely arbitrary because the cable is constrained to remain flat. Third, to further minimize discontinuities and provide smoother transitions, additional segments were added between randomly generated sections.
 The procedure for generating the random cable is illustrated in Figure 2. The overall cable can be thought of as being confined to a rectangular box. It tends in the z direction and is not allowed to double back on itself. Its length is divided into sections, and at each section, a random (x, y) position is generated that the cable passes through, as shown in Figure 2. From these random positions, the orientation of the cable at that position and at positions in between can be determined as shown by the dark arrows in Figure 2. The calculations are partly determined by the random (x, y) positions and also by the constraint that the cable is flat. To reduce discontinuities, each section is further divided into smaller sections, and the cable orientation within these smaller sections is interpolated from the orientation at the random points.
 For this work, the dimensions of the rectangular box shown in Figure 2 were 4 cm in the x direction and 6 cm in the y direction. Figures 3 and 4 show histograms of the random positions in the x and y directions. The envelope of the histogram gives the PDF. From Figures 3 and 4 it can be seen that a PDF was used that tends to keep the cable centered in the rectangular box.
 At each of the randomly generated points, an additional +π or −π rotation about the cable axis can be added without changing the overall lay of the cable. Therefore in addition to the x and y position, random twists were added during the cable generation process. A PDF was chosen such that there was a 50% chance of no extra twist, a 25% chance of an additional −π twist and a 25% chance of an additional +π twist as shown by the following PDF:
 The particular distribution shown in (17) is for example purposes only. For a real application, a detailed study should be performed to determine the actual distribution.
 The orientation of the cable at the terminations was not random. In (4) and (5), it can be seen that there is a source term due to the incident electric fields transverse to the cable at the terminations. In this work, it was decided not to consider this term and instead concentrate on the sources that arise because of the incident fields along the cable length. This was achieved by holding the orientation of the terminations constant and in such a way that there were no electric fields transverse to them.
 For this example the incident field was a 1 V/m plane wave traveling in the −x direction and with an electric field in the +z direction. The source terms were found using (12) and (13). To use (12) and (13) the following are needed: the incident electric field longitudinal to each section of the cable, the propagation constant longitudinal to each section of the cable, and the propagation constant transverse to each section of the cable. Using the source terms, the MTL equations were then solved, giving the voltages and currents at the source and load ends of the cable.
 In the course of this work numerous parameters were altered to determine their affect on the results. Some of these parameters included the termination values, cable length, the number of wires, and homogeneous and nonhomogeneous media. In addition, the amount of random variation in the cable was varied by altering the number of sections the cable was divided into and by the number of twists. Here, only the results of a specific example, which are indicative of others, are shown. The conditions were as follows: the cable length was 10 m, the number of wires was 20, and all wires were terminated with 50 Ω to ground. For purposes of generating random positions, the cable was divided into 32 sections. However, there were a total of 330 sections due to those added to reduce discontinuities as was explained previously. The length of each segment was 3 cm. That is much smaller than the minimum wavelength considered which was 30 cm at f = 109 Hz.
 The simulations performed generate the steady state frequency response of the voltages and currents at the terminations on both the source and load ends of the cable. Therefore for an n + 1 wire MTL, there are 4n quantities that are solved for. It would be difficult to show such a significant amount of data here. Therefore only the magnitude of the voltage on the third wire from the reference wire at the source termination is shown. This choice was arbitrary, and is indicative of others. The frequency range chosen was from 104–109 Hz. This covers a low-frequency range where the cable is considered electrically short; meaning it is much smaller than the excitation frequency. It also covers a high-frequency range where the cable is considered electrically long.
Figure 5 shows the frequency response of a cable whose orientation was randomly generated 250 times. The results of each trial are plotted simultaneously. From Figure 5 some features of the response that can be seen include the typical 20 dB/decade response in the electrically short region and the resonances in the electrically long region. Such a plot gives a visual indication and estimation of the distribution at each frequency. For example, looking at Figure 5 at f = 104 Hz, it can be seen that the majority of the response falls between −80 and −60 dBmV, but the exact amount cannot be determined. For an accurate result the CDF, which gives the probability that a random variable is less than or equal to an outcome, is needed.
Figure 6 shows the CDF at f = 104 Hz. Looking at Figure 6, it can be seen that there is a 91% probability that the response is less than −60 dBmV and a 14% probability that it is less than −80 dBmV. Therefore there is a (91% − 14%) = 77% probability that it is in the range between −80 and −60 dBmV. The CDF of Figure 6 was generated using 5 · 104 samples, that is to say 5 · 104 random cable realizations. Figure 7 shows another example of a CDF which was taken at f = 108 Hz, which is in the electrically long region of the response.
 Another way to view the distribution is the PDF, which is the derivative of the CDF. Here the CDF is not an analytical function and therefore the PDF cannot be found by taking the derivative. Instead, the PDF is found by generating a histogram of the response. The histogram divides the possible responses into bins and then determines the number of times the response falls into each bin. The envelope of the histogram gives the PDF. The accuracy depends on the size of the bins and the number of responses generated. Figure 8 shows the histogram of the response at f = 104 Hz. The histogram is based on 5 · 104 samples and each bin holds a 1 dBmV range. From the PDF, the probability of a given occurrence can be determined. For example, it can be seen that there is a 5.5% chance of a response of −65 dBmV occurring. Figure 9 shows another example of a PDF which like the CDF of Figure 7 was taken in the electrically long region at f = 108 Hz.
 As previously explained only the response on a particular wire, the third from ground, has been shown. This was solely done for clarity of explanation but was not a necessity. For example it may be instructive to examine the results of various wires to see if position dependence is found. Figure 10 is similar to Figure 6 except is shows the CDF for all wires. From Figure 10 it can be seen that all wires have a similar CDF though shifted in value with the wires farthest from the ground wire having the largest response as would be expected from the problem geometry. Previously it was also explained that this study was performed in the frequency range from 104–109 Hz. However, only results from two specific frequencies have been shown. This was done to provide a clear illustrative example. However, it may be desirable to examine the response as a function of frequency as opposed to at a single frequency. Figure 11 shows a surface plot illustrating the CDF of a single wire, again the third from ground, as a function of frequency. Figure 11 is reminiscent of Figures 5 and 6 but giving a more complete view of how the line is responding across frequency.
 In this paper, a model was shown for calculating the terminal response on a cable excited by an incident electromagnetic field where the orientation of the cable is random with respect to that field. The model is based on transmission line theory, which offers a computational advantage over other potential methods such as full wave solvers. It can be applied to cables with any number of conductors as well as uniform, nonuniform, homogeneous, nonhomogeneous, lossy, or lossless lines. Random cable bundles are modeled by segmenting them into a number of smaller sections and allowing the orientation of the sections to vary randomly. In this way, an overall random cable is obtained from which the response due to an external field can be found. By generating a large number of random configurations, the statistical nature of the response is obtained.
 An example was shown by applying this model to a ribbon cable. First the random cable was generated and then the CDF and PDF of the response due to an incident external field were found. The ribbon cable example was that of a uniform line, but it was also shown how the more general nonuniform line could be modeled. For the NMTL case, the line is again divided into segments. However, the PUL parameters for each segment can vary as well as their orientation with respect to an external field.
 The ribbon cable example was that of isolated wires. However, wires over a ground plane could also be modeled. It would not be possible to model such a configuration using the rotated sections of the ribbon cable example because the ground plane is the return conductor and its orientation does not change. Such a cable is still modeled by breaking it into discrete sections; however, those sections would resemble a staircase with each section having the same orientation. With this method, random wire heights above the ground, wire spacing, and twists can be captured.
 This work was supported by the U.S. Department of Defense under MURI grant F49620-01-1-0436.