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A FAST STROUD-BASED COLLOCATION METHOD FOR STATISTICALLY CHARACTERIZING EMI/EMC PHENOMENA ON COMPLEX PLATFORMS

Abdulkadir C. Yücel1, Hakan Bağcı1, Jan S. Hesthaven2, and Eric Michielssen 1

(1) Department of Electrical Engineering and Computer Science University of Michigan at Ann Arbor

Ann Arbor, MI 48109, USA

(2) Division of Applied Mathematics Brown University

Providence, RI 02912, USA

Abstract

A fast stochastic collocation method for statistically characterizing electromagnetic interference and compatibility (EMI/EMC) phenomena on electrically large and loaded platforms is presented. Uncertainties in electromagnetic excitations and/or system geometry and configuration are parametrized in terms of random variables having normal or beta probability density functions. A fast time domain integral equation-based field-cable-circuit simulator is used to perform deterministic EMI/EMC simulations for excitations and/or system geometries and configurations specified by Stroud integration rules. Outputs of these simulations then are processed to compute averages and standard deviations of pertinent observables. The proposed Stroud-based collocation method requires far fewer deterministic simulations than Monte-Carlo or tensor-product integrators. To demonstrate the accuracy, efficiency, and practicality of the proposed method it is used to statistically characterize coupled voltages at the feed pins of cable-interconnected and shielded PC cards as well as the terminals of cables located in the bay of an airplane cockpit.

Keywords—electromagnetic interference and compatibility, stochastic collocation, Stroud integration rules, electromagnetic coupling, time domain integral equations, hybrid simulators, fast solvers

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I. Introduction

The proper functioning of electronic communication, navigation, and sensing systems on modern vehicles

often is threatened by internally or externally generated electromagnetic interference [1-3]. The

characterization of these systems’ vulnerability to interference is complicated by the fact that

electromagnetic fields inside electrically large platforms (e.g., as vehicle shells) often behave quasi-

chaotically. Small perturbations in the excitation (e.g., as the angle of arrival and polarization of an

incident plane wave, the waveform of a source, etc.) and the system geometry and configuration (e.g., as

manufacturing or installation ambiguities in the routing of cables, the placement of electronic systems,

and the makeup of circuits and lumped element devices, etc.) may dramatically impact the

electromagnetic interference and compatibility (EMI/EMC) performance of the overall system.

For electromagnetic simulators to be truly useful in the characterization of EMI/EMC phenomena on

realistic structures, they must be computationally efficient and offer multiscale simulation capabilities to

permit the analysis of electromagnetic phenomena on electrically large platforms loaded with

(sub)wavelength-scale systems. Furthermore, they must be able to accurately account for the possible

perturbations (i.e. uncertainties) in the electromagnetic excitation and the system geometry and

configuration.

The time-domain integral-equation (TDIE)-based simulator described in [4-8] has successfully addressed

the first of the above mentioned two simulation challenges: It permits the efficient EMI/EMC analysis of

deterministically configured, electrically large, multiscale, and loaded platforms. This simulator, which

derives its computational efficiency from fast Fourier transform (FFT)-based acceleration engines [9, 10]

and parallelization [11, 12], hybridizes three distinct solvers [4-8]; (i) a field solver computes fields on the

electrically large platform, and any antennas, shielding enclosures, boards, and shields of cables present;

(ii) a cable solver computes transmission-line voltages and currents on cables interconnecting electronic

(sub)systems; (iii) and a circuit solver computes node voltages on lumped element circuits that model

electrically small components.

This work presents an extension to the above TDIE-based simulator aimed at statistically characterizing

EMI/EMC phenomena on electrically large, multiscale, and loaded platforms with uncertain system

geometries and configurations and subject to variable electromagnetic excitations. The proposed

extension leverages the Stroud-based stochastic collocation methods described in [13-18]. To compute the

multidimensional integral pertinent to the statistical characterization of an observable (e.g. computation of

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the average or standard deviation of a voltage somewhere deep into an electronic system), the proposed

method calls for the evaluation of that observable (using the above deterministic EMI/EMC simulator) at

collocation points (i.e. for specific excitations and/or system geometries and configurations) derived using

so-called Stroud integration rules. Assuming the dimension of the space parametrizing all uncertainties is

dofN , Stroud-2 (S-2) and Stroud-3 (S-3) integration rules require only dof 1N + or dof2N collocation

(integration) points [13-16, 18] to evaluate the dofN -dimensional integrals defining the average and the

standard deviation of the observable to be characterized; for S-2/S-3 rule to deliver accurate results, the

observable and its square should be adequately described by a second/third order polynomial in the dofN -

dimensional parameter space describing the uncertainties. The selection of the collocation points is

influenced by the random variables’ (assumed) probability density functions (p.d.f.s), which may be

normal or beta.

The Stroud integration rules used here are far more efficient than Monte Carlo or tensor-product ones.

Indeed, Monte Carlo integrators [16, 19] call for the evaluation of the observable at (quasi-)randomly

selected points in the dofN -dimensional space of parameters describing uncertainties; the integration

accuracy is inversely proportional to the square of the number of points selected. Even though their

implementation is straightforward, Monte Carlo integrators often require too many deterministic function

evaluations to yield reasonably accurate data. Of course, the accuracy of the multidimensional integral’s

evaluation can be improved using tensor-product integration rules [16, 20]. Unfortunately, if an n point

one-dimensional (1-D) integration rule is used along each dimension of an dofN - dimensional integral, the

tensor-product integrator requires dofNn deterministic function evaluations. The computational cost

associated with these evaluations often is prohibitive even for moderate n and dofN , and stands in stark

contrast with that incurred when using S-2 and S-3 rules, which call for dof 1N + and dof2N number of

function evaluations, respectively. The accuracy of the Stroud-based collocation method adopted here is

comparable to that of the stochastic Galerkin method, where the solution in the space of parameters

describing uncertainties is represented using a Chaos expansion and the resulting equations are satisfied in

a Galerkin sense [17, 18]. Even though this approach makes the stochastic Galerkin method slightly more

accurate than the Stroud-based collocation methods for some problems, it requires the solution of fully

coupled systems, i.e., the development of new solvers. Stroud-based collocation methods are preferred in

this work since they are straightforward to integrate into existing electromagnetics simulators.

II. Formulation

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This section details the proposed method for statistically characterizing EMC/EMI phenomena on

electrically large, multiscale, and loaded platforms with uncertain system geometries and configurations

and subject to variable electromagnetic excitation. Subsection II-A describes the TDIE-based simulator

that is used to perform the deterministic function evaluations needed for the stochastic collocation

method. Subsection II-B describes the model for the parameter space describing uncertainties and details

the stochastic collocation method derived from S-2 and S-3 integration rules for its discretization.

A. Description of the TDIE-based Hybrid Electromagnetic Simulator

This section briefly describes the deterministic TDIE-based simulator; it details its three solver

components (TDIE-based field, TDIE-based cable, and modified nodal analysis (MNA)-based circuit

solvers) as well as their rigorous coupling at the cable shields and the circuit terminals.

• TDIE-based field solver: The surface of the electrically large platforms, antennas, shielding

enclosures, boards, and cable shields (exterior structure) is represented by perfect electrically

conducting bodies and thin wires that may be attached to the bodies by means of junctions. The field

solver numerically solves a three-dimensional (3-D) TDIE enforced on the exterior structure’s surface

for induced electric current density: First, the current density on exterior structure’s surface is

expanded using EMN spatial and tN temporal basis functions. Spatial basis function set consists of

surface, wire, and junction basis functions. Then, this expansion is inserted into the 3-D TDIE and the

resulting equation is tested using Galerkin’s method in space and point matching in time. This results

in a EM EMN N× linear system of equations, which is solved using the well-known marching-on-in-

time (MOT) technique. The TDIE-based field solver’s computational bottleneck is the computation of

the fields due to “past” currents, which scales as ( )2t EMO N N . The computational complexity of this

operation is reduced to ( )2t c c EMlogO N N N P using the time-domain adaptive integral method (TD-

AIM) and parallelization [10]. Here, cN and EMP are the numbers nodes of an auxiliary 3-D uniform

Cartesian grid that encloses the exterior structure and processors assigned to the field solver,

respectively.

• TDIE-based cable solver: Under transverse electromagnetic (TEM)-like propagation assumption,

guided fields along coaxial cables are represented by voltage and current waves satisfying the well-

known transmission-line equations [21, 22]. The lossy and dispersive nature of the guided-wave

propagation along the transmission lines is accounted for by using the pertinent transmission-line

Green function. The cable solver numerically solves a 1-D TDIE, which is obtained by enforcing the

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boundary conditions relating voltages and currents at the cable terminals, for current wave variables

[23, 24]. First, CBL2N wave variables are expanded using tN temporal basis functions ( CBLN is the

total number of coaxial cables). Then, this expansion is inserted into the 1-D TDIE and the resulting

equation is tested using point matching in time. This results in a CBL CBL2 2N N× linear system of

equations, which is solved using the standard MOT recipe. The computational bottleneck of the

MOT-based solution is the evaluation of the voltages and currents, which requires the convolution of

transmission-line Green functions with the past voltages and currents. The computational complexity

of this computation, which scales as ( )2tO N is reduced to ( )2logt tO N N using the FFT-based

algorithm of [9].

• MNA-based circuit solver: Electrically small components are represented by equivalent (possibly

nonlinear) lumped circuits. The circuit solver numerically solves Kirchhoff’s equations for the node

voltages and voltage-source currents on these lumped circuits. Enforcing Kirchhoff’s equations for

CKTN total number of non-ground nodes via MNA yields a CKT CKTN N× (non)linear system of

equations, which is solved at every time step [11].

Combining the systems of equations pertinent to each solver (while carefully accounting for the field

interactions at circuit and cable terminals, and along the cable shields as explained in [4] in detail), results

in a ( ) ( )EM CKT CBL EM CKT CBL2 2N N N N N N+ + × + + coupled system of equations (Fig. 1), which is solved

simultaneously for all field, cable, and circuit solvers’ unknowns. The first row of the coupled system

features the EM EMN N× MOT system; EMlI is the vector of unknown current coefficients, EM

-l l′Z are MOT

impedance matrices, eilV is the excitation vector, e

lV is the vector of scattered fields due to past currents, ecC is the matrix that selects circuit terminals connected to the exterior structure, and ec

lV is the vector

that represents coupling from past circuit terminal voltages. The second row of the coupled system

features the CKT CKTN N× MNA system; CKTlV is the vector of unknown node voltages and voltage-source

currents, CKTY is the MNA admittance matrix, ( )CKT,nl CKTllI V is the vector representing nonlinear branch

equations, cilI is the excitation vector, ec†C is the matrix that selects spatial basis functions connected to

circuit terminals, and cte cte/ lC I and ch ch/ lC I are the matrices/vectors that represent coupling from

present/past external fields (through cables) and guided fields (directly), respectively. The third row of the

coupled system is the CBL CBLN N× MOT system; CBLlI is the vector of unknown wave variable

coefficients, il l′−G are cable-propagation matrices, tcC is the matrix that selects circuit terminals

connected to cable terminals, and te te/ lC V is the matrix that represents coupling from present/past

external fields.

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The coupled system of equations is solved simultaneously for EMN field, CKTN circuit, and CBL2N cable

unknowns at each time step using Newton-Raphson algorithm of [11]. The solution cost is dominated by

the computations of right-hand-side vectors hlV , cte

lI , chlI , te

lV and elV , which are accelerated by

FFT-based algorithms [9] and TD-AIM [10], respectively. Further acceleration is achieved by distributing

the workload among P processors in line with the approach in [11]: The first EM 1P P= − processors are

assigned to the field-solver and the thP processor is assigned to the circuit and cable solvers, respectively.

This approach maintains effective load balancing as the field solver requires far more computations than

the circuit and cable solvers ( EM CKT CBLN N N+ ) [12].

B. Stochastic Models and Stroud-Based Collocation Methods

I. Stochastic Models

Let dof dof11 2, ,..., ,N Nx x x x−⎡ ⎤= ⎣ ⎦x represent an dofN -vector that parametrizes uncertainties in excitation (e.g.,

the angle of arrival and polarization of an incident plane wave, the waveform of a source, etc.) and/or

system geometry and configuration (e.g. the positions of cables and electronic systems, the values of

lumped circuit elements, devices descriptions, resistive and surface impedances, etc.). Furthermore, let

( )W x denote the assumed multidimensional p.d.f. of x . Here, ( )W x is expressed as a product of non-

negative 1-D p.d.f.s ( )iw x , dof1,..., ,i N= i.e.

( ) ( )dof

1

Ni

i

W w x=

=∏x . (1)

The random variables ix , dof1,...,i N= , are assumed to be either normally distributed with mean iμ and

standard deviation iσ , or beta distributed defined in the range ,i ia b⎡ ⎤⎣ ⎦ with exponential parameters iα

and iβ . Table I presents explicit expressions for normal and beta p.d.f.s; note that the uniform p.d.f. is a

special case of the beta distribution with 0i iα β= = . The vector x takes on nonzero values in the

domains ( ) dof, ND = −∞ ∞ or ( )dof dof1 1, ,..., ,N ND a b a b⎡ ⎤⎡ ⎤= ⎣ ⎦ ⎣ ⎦ depending on whether the ix are normal or

beta distributed random variables. Even though integration rules for both normal and beta p.d.f.s are

implemented in this work, modeling uncertainties with normal distribution typically is not recommended

because of inefficiencies incurred with the modeling of the distribution’s infinite tail [17]. In practical

applications, normal distributions are often well-approximated by beta distributions [17].

Assume one is interested in statistically characterizing the voltage coupled onto the port of an electronic

system as a function of frequency. This voltage’s average (expectation value) and standard deviation (for

a given frequency) are expressed as

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( ) ( ) ( )DV W dE V⎡ ⎤

⎣ ⎦ = ∫ x x xx , (2)

( ) ( ) ( )( ) ( )2

stdD

V E V W dV⎡ ⎤ ⎡ ⎤− ⎣ ⎦⎣ ⎦ = ∫ x x x xx . (3)

Here, ( )V x is the Fourier transform of the transient coupled voltage ( )V x , which can be efficiently

evaluated by the above described deterministic simulator for any realizable x . The question arises as to

how to efficiently evaluate the dofN -dimensional integrals (2)-(3). Note that if the observable of interest

was not a frequency domain voltage, but a frequency domain current or field, or for that matter a transient

voltage, current, or field, integrals similar to (2)-(3) would still need to be computed.

II. Stroud-Based Collocation Methods

The S-2 and S-3 integration rules originally proposed by Stroud [13] and recently extended to allow for

the approximation of multidimensional integrals with arbitrary weighting functions in [15] allow for the

efficient evaluation of (2)-(3) provided ( )V x and ( )2

V⎡ ⎤⎣ ⎦x are (or can be well-approximated by) second

or third order polynomials in x . These rules approximate these integrals as

( ) ( ) ( )x

1

.M

i iiD

f W d w V=

≅ ∑∫ x x x x (4)

Here, ( ) ( )f V=x x or ( ) ( ) ( )( )2f V E V⎡ ⎤= − ⎣ ⎦x x x depending on whether (2) or (3) is being

approximated, M is the number of integration points, and ix and xiw , 1,...,i M= , are integration points

and weights. Expressions for the latter quantities are given in Table II for normal and beta p.d.f.s; the

expressions provided were derived from those in [15], which assumed standard normal distributions

( 0iμ = , 1iσ = ) and normalized beta distributions ( [ ], 1,1i ia b⎡ ⎤ = −⎣ ⎦ ) via a linear mapping. In practice, the

approximation in (4) will converge to the exact integral provided the integration domain D is small

enough. S-2 and S-3 integration rules require dof 1M N= + and dof2M N= number of integration points,

respectively. This renders Stroud integrators much more efficient when compared to Monte Carlo or

tensor-product ones.

Tensor-product integrators [16, 20] treat a multidimensional integral as a nested sequence of 1-D integrals

that are evaluated using 1-D integration rules, typically Gaussian ones [25] (Gauss-Hermite for random

variables with normal p.d.f [25] and Gauss-Jacobi for random variables with beta p.d.f [26]) as they

provide maximum accuracy for a given number of integration points. Unfortunately, when using an n -

point 1-D integration rule for each dimension of an dofN - dimensional integral, a tensor-product

integrator calls for dofNn function evaluations. Therefore, even for EMI/EMC scenarios with low-

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dimensional parameter spaces, the use of tensor-product integrator is not possible. In the next section,

Gauss-Hermite and Gauss-Jacobi based tensor-product integrators are implemented to compute the

multidimensional integrals in (2) and (3); the number of deterministic simulations required by the Stroud

integrator is small compared to that of the tensor-product integrator for several practical examples where

both integrators have the same level of accuracy. Sparse grid methods strike an interesting compromise

between the efficiency of Stroud integrators and the accuracy of tensor-product integrators. These

methods are discussed at length in [14]; it is the authors’ opinion that for many EMI/EMC studies Stroud

integrators yield sufficient accuracy, rendering sparse grid methods less relevant.

Monte Carlo integrators [16, 19] can also be used to compute the multidimensional integral (4). Monte

Carlo integrators first select random sampling points kx , 1, ,k M= … , that have the same p.d.f. as ( )W x ,

then approximate the integral as the average of all ( )kf x , 1, ,k M= … . Monte Carlo integrators are

easily implemented, however their accuracy scales only as ( )1O M [16, 19]. Even though both t Monte

Carlo and Stroud integrators call for the evaluation of the observable for specific excitations and system

realizations, Monte Carlo integrators are highly inefficient compared to Stroud ones, especially for small

dofN (i.e. dofM N has to be satisfied to obtain same level of accuracy as the Stroud integrator). This

prohibits using Monte Carlo integrator for many realistic stochastic EMI/EMC scenarios because of the

lengthy CPU times required for many deterministic EMI/EMC simulations. The reason for this drastic

increase in the efficiency can be explained as follows. All samples in the Monte Carlo integrator are most

often given the same weight unlike the Stroud integrator where locations of the samples (integration

points) in the parameter space and their weights are chosen carefully. However, to be able obtain the

Stroud integration points and weights, one needs to assume that random variables can be approximated

using low-order (smooth) polynomials. In problems, where this assumption is not satisfied, a Monte Carlo

integrator would often be superior. Fortunately, in many EMI/EMC problems, uncertainties can be

modeled by smoothly varying functions, thus ensuring the efficiency of the Stroud-based stochastic

collocation method as shown in numerical results section.

III. Numerical Results

This section presents numerical examples that demonstrate the accuracy, efficiency, and practicality of

the proposed approach via its application to the statistical characterization of plane wave coupling into

suspended coaxial cables (Subsection III-A), onto the feed pins of shielded and cable-interconnected PC

cards (Subsection III-B), and into the coaxial cables situated in the bay of an airplane cockpit (Subsection

III-C).

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A. Plane Wave Coupling into Suspended RG-58 Coaxial Cables

The proposed approach is used to estimate the average and standard deviation of voltages coupled into the

terminals of an RG-58 coaxial cable excited by a plane wave. The cable has polyethylene dielectric

filling, wave speed CBL 00.78c c= with 0c the free-space speed of light, outer shield radius

1.524 mmoa = , inner shield radius 1.397 mmia = , and inner conductor radius 0.180 mma = . The cable

shield’s transfer impedance is approximated as

( ) ( )( ) ( )( )i0 o i o i

ˆ 1 / / sinh 1 / 2 aT f R j a a j a a j fLδ δ π= ⎡ + − ⎤ ⎡ + − ⎤ +⎣ ⎦ ⎣ ⎦ , where 1 fδ π σμ= is the skin

depth, 75.0 10 S/mσ = × is the conductivity, 0μ μ= is the free-space permeability, 1.0 nH/maL = , and

0 14.3 mΩ/mR = [27]. Due to the high optical coverage of the cable, the shield’s transfer admittance is

negligible [27]. The cable resides in free-space, is flexible and uniform, and suspended at fixed nodes that

lie on the x -axis; the sagging between any two nodes is characterized using the catenary curve equation

[28]. The cable is terminated by two resistors and illuminated by a plane wave propagating in the

( ) ( )ˆ ˆ ˆcos sinφ φ= +k x y direction with 1 V/mEφ = . Two different scenarios are simulated.

I. RG-58 Cable Suspended at Three Points

In the first scenario, an RG-58 coaxial cable of length 6 m is suspended at three nodes that are spaced

2 md = apart [Fig. 2 (a)]. In this example, five parameters characterize the uncertainty ( dof 5N = ): the

values of the terminating resistors, 1R and 2R , the maximum cable sag between the fixed nodes, 1h and

2h , and the plane wave’s angle of arrival, φ . All five variables are assumed normally distributed. Five

sets of simulations ( 1,...,5i = ) are performed in which these variables’ standard deviations are

geometrically increased while their means are kept constant (Table III). The average and standard

deviation of the coupled voltages at node 1 are computed at 100 MHzf = using S-2/S-3 integrators, a

tensor-product integrator that uses an 5 -point Gauss-Hermite integration rule, and a Monte Carlo

integrator; results obtained using the Gauss-Hermite integrator were verified to be accurate to 8 and 5

digits for the first and last simulations, respectively (by comparison to results obtained using even higher-

order Gauss-Hermite integrators). Fig. 2 (b) shows the relative error between the averages computed

using the S-2/S-3 ( 2,3k = ) integrator (S) and the tensor-product integrator (P),

( ) ( )

( )

S, P1 1E,Re,

P1

Re Re, 1,...5, 2,3

Re

ki ik

i

i

E V E Verr i k

E V

⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦= = =⎡ ⎤⎣ ⎦

x x

x, (5)

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( ) ( )

( )

S, P1 1E,Im,

P1

Im Im, 1,...,5, 2,3

Im

ki ik

i

i

E V E Verr i k

E V

⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦= = =⎡ ⎤⎣ ⎦

x x

x. (6)

Similarly, Fig. 2 (c) shows the relative difference between the standard deviations computed by the S-2/S-

3 ( 2,3k = ) integrator (S) and the tensor-product integrator (P),

( ) ( )

( )

S, P1 1std,Re,

P1

std Re std Re, 1,...5, 2,3

std Re

ki ik

ii

V Verr i k

V

⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦= = =⎡ ⎤⎣ ⎦

x x

x, (7)

( ) ( )

( )

S, P1 1std,Im,

P1

std Im std Im, 1,...,5, 2,3

std Im

ki ik

i

i

V Verr i k

V

⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦= = =⎡ ⎤⎣ ⎦

x x

x. (8)

In (5)-(8), ( )1V x is the coupled voltage at node 1, [ ]S,2 .iE / [ ]S,2std .i , [ ]S,3 .iE / [ ]S,3std .i , and [ ]P .iE / [ ]Pstd .i

represent the averages/standard deviations computed by the S-2/S-3 and tensor-product integrators for the thi set of simulations, respectively. As expected, relative errors plotted in Figs. 2 (b)-(c) increase as the

standard deviations iσ increase (because the effective domain of integration enlarges). Additionally, one

can say that for a given set, the error in averages is one-two orders of magnitude smaller than the error in

standard deviation; this is expected since the integral needed for average involves only the coupled

voltage ( )1V x , rather than ( )2

1V⎡ ⎤⎣ ⎦x , which is integrated for computing the standard deviation (i.e. the

function being integrated for computing the average is smoother than the one integrated for computing the

standard deviation). For a more detailed comparison of the S-2/S-3, the Gauss-Hermite, and the Monte

Carlo integrators, Table IV presents the results computed for the third set of simulations by these

integrators [Third simulation is the one in the middle for plots presented in Figs 2 (b) and (c)]. In Table

IV [ ]M3 .E / [ ]M

3std . represents the average/standard deviation computed by the Monte Carlo simulation for

the third set of simulations. To obtain the results given in the Table IV, the S-2/S-3, Gauss-Hermite, and

Monte Carlo integrators required 6 , 10 , 3125 , and 5000 deterministic EMI/EMC simulations,

respectively. As expected, for both the average and the standard deviation computations, the S-3

integrator gives more accurate results than the S-2 integrator (assuming that the most accurate results are

obtained by the Gauss-Hermite integrator). Additionally, both S-2 and S-3 integrators are approximately

one digit more accurate than 5000-point Monte Carlo integrator. For the Monte Carlo integrator to yield

roughly the same accuracy as the S-2/S-3 integrators, it would require approximately 100 5000× points.

It is clear from the results presented in this section that, for a stochastic problem this size, Stroud

integrators are the most efficient and can provide one-two digits of accuracy even for relatively large

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integration domains. For the sake of completeness, Figs. 2 (d) and (e) show ( ) S,33 1ReE V⎡ ⎤⎣ ⎦x and

( ) S,33 1ImE V⎡ ⎤⎣ ⎦x vs. frequency with error bars obtained from ( ) S,3

3 1std Re V⎡ ⎤⎣ ⎦x and ( ) S,3

3 1std Im V⎡ ⎤⎣ ⎦x ,

respectively. As expected, the standard deviation of the coupled voltage increases near the cable’s

resonant frequencies, where the cable’s response is more sensitive to perturbations in the excitation and

the configuration of the cable.

II. RG-58 Cable Suspended at Eleven Points

In the second scenario, the RG-58 coaxial cable of length 18.63 m is suspended at 11 nodes that are

spaced 1 md = apart [Fig. 3 (a)]. 13 parameters characterize the uncertainty ( dof 13N = ): the values of

the terminating resistors 1R and 2R , the maximum cable sag between nodes ih , 1, ,10i = … , and the angle

of arrival of excitation φ [Fig. 3 (a)]. The variables 1R , 2R , ih , 1, ,10i = … , and φ are assumed

uniformly distributed in the ranges [ ]45 55 − Ω , [ ]45 55 − Ω , [ ]0.5 0.7 m− , and [ ]225 235− ° ,

respectively. Next, results obtained using S-3 and Monte Carlo integrators are compared in Table V. In

Table V, ( )1V x is the coupled voltage at node 1 at 0.9 GHzf = , and [ ]S,3 .E / [ ]S,3std . and

[ ]M .E / [ ]Mstd . are the averages/standard deviations computed by the S-3 and Monte Carlo integrators,

respectively. The relative differences between ( ) S,31std Re V⎡ ⎤

⎣ ⎦x and ( ) M1std Im V⎡ ⎤

⎣ ⎦x , and

( ) S,31std Im V⎡ ⎤

⎣ ⎦x and ( ) M1std Re V⎡ ⎤

⎣ ⎦x are 1.43% and 4.09% , respectively. The relative differences

between ( ) S,31ReE V⎡ ⎤

⎣ ⎦x and ( ) M1ReE V⎡ ⎤

⎣ ⎦x , and ( ) S,31ImE V⎡ ⎤

⎣ ⎦x and ( ) M1ImE V⎡ ⎤

⎣ ⎦x are very

large for the simple reason these quantities are (likely) approximations to zero. The averages of the real

and imaginary parts of the coupled voltage are expected to vanish because of phase cancellations that

occur when the number of and/or variation in the parameters quantifying the uncertainties are large [29,

30]. Under these conditions, the p.d.f.s of the real and imaginary parts of the coupled voltage and its

absolute value are expected to behave as normal and Rayleigh p.d.f.s, respectively [29, 30]. These facts

are verified by the histograms obtained using the Monte Carlo integrator [Figs. 3 (b)-(d)]. For this

example, the S-3 and the Monte Carlo integrators required 26 and 5000 deterministic EMI/EMC

simulations, respectively.

B. PC Cards in Shielding Enclosures

Next, the proposed approach is used to statistically characterize coupled voltages at terminals of an RG-

58 coaxial cable connecting two PC cards located inside shielding enclosures that are subject to excitation

by a plane wave [Fig. 4 (a)]. The shielding enclosures are identical; both contain a mother board and two

daughter cards. The daughter card closest to the back of the box ( st1 card) and the other one ( nd2 card) are

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connected to the mother board with eight pins and one pin, respectively (See [4] for a more detailed

description the cards and the pins). An RG-58 coaxial cable of length 70 cm but otherwise identical to

the one used in Subsection III-A connects to the pins feeding the nd2 cards. Two resistors, which might

potentially model the resistance of cable connectors, are also connected to feed pins. The structure is

illuminated by a plane wave propagating in ( ) ( )ˆ ˆ ˆsin cosθ θ= − −k x z direction with 1 V/mEθ = . Three

parameters characterize the uncertainty ( dof 3N = ): the values of the terminating resistors 1R and 2R , and

the plane wave’s angle of arrival θ ; all are assumed beta distributed with 1α β= = in the ranges

[ ]48 52 − Ω , [ ]48 52 − Ω , and [ ]110 115− ° , respectively. S-3 and a tensor-product integrators are used

to compute averages and standard deviations of the voltage ( )1V x coupled onto node 1 at 0.9 GHzf = ;

the tensor-product integrator uses a 5-point Gauss-Jacobi integration rule. The results are compared in

Table VI, where [ ]S,3 .E / [ ]S,3std . , and [ ]P .E / [ ]Pstd . represent the averages/standard deviations computed

by the S-3 and the Gauss-Jacobi integrators, respectively. The averages and standard deviations computed

by integrators match up to sixth and fourth digits, respectively. For this example, the S-3 and Gauss-

Jacobi integrators required 6 and 125 deterministic EMI/EMC simulations, respectively. For the sake of

completeness, Figs. 4 (b) and (c) show ( ) S,31Re E V⎡ ⎤⎣ ⎦x and ( ) S,3

1Im E V⎡ ⎤⎣ ⎦x vs. frequency with error

bars obtained from ( ) S,31Re std V⎡ ⎤⎣ ⎦x and ( ) S,3

1Im std V⎡ ⎤⎣ ⎦x , respectively. As expected, the standard

deviation of the coupled voltage increases near the resonant frequencies of the system.

C. RG-58 Coaxial Cables in Loaded Cockpit

Finally, the proposed approach is used to statistically characterize voltages coupled onto the terminals of

RG-58 coaxial cables situated inside the bay of an airplane cockpit under plane wave excitation [Fig. 5

(a)]. The cockpit is loaded with three shielding enclosures interconnected by RG-58 coaxial cables, as

well as nine additional floating RG-58 coaxial cables and two seats [Fig.5(a)]. Aside from their length, all

RG-58 coaxial cables are identical to those of Subsection III-A. The cables are located in a partially

shielded compartment under the cockpit floor in three layers [Fig. 5 (b)]. Cables 2 ,C 3 ,C and 6C are

1.375 m long, cables 1,C 4 ,C 5 ,C and 7C are 1 m long, and cables 8C and 9C are 0.575 m long.

Cables 1,C 6 ,C and 8C are terminated using resistors with values 1R and 2R , 3R and 4 R , and 5R

and 6 ,R respectively; 50 ,iR = Ω 1,...,6i = . The surfaces of the seats are modeled as resistive surfaces

with 377-Ω surface impedance. The structure is illuminated by a plane wave propagating in the

( ) ( )ˆ ˆ ˆcos sinθ θ= − −k x y direction with 1000 V/m.Eθ = Nine parameters characterize the uncertainty

( dof 9N = ): the y coordinates of cables 1C , 4C , 5C , 7C , 8C , and 9C [Fig. 5 (b)], the y coordinates of

the seat base plane, and the plane wave’s angle of arrival θ . All nine random variables are beta

13

distributed with 0i iα β= = , 1,...,9i = , in the range [ ]i ia b ; ia and ib are given in Table VII. The S-3

integrator is used to compute ( )S,3iE V⎡ ⎤

⎣ ⎦x and ( )S,3std iV⎡ ⎤⎣ ⎦x , 1,...,6i = , the average and standard

deviation of the absolute value of the coupled voltages on resistors ,iR 1,...,6i = , at 0.9 GHzf = (Table

VIII). For this problem, only the S-3 integrator, which required 18 deterministic simulations, was used

because the time required for a single deterministic simulation was around 2-3 hours on 32 processors,

and a tensor-product integrator with 2 points in each dimension and Monte Carlo integrator with a couple

of digits accuracy would require 92 512= and at least a few thousands simulations, respectively.

IV. Conclusion

A fast Stroud-based stochastic collocation method for statistically characterizing EMI/EMC phenomena

on electrically large and loaded platforms is presented. Uncertainties in electromagnetic excitations and/or

system geometries and configurations are parametrized in terms of variables having normal or beta p.d.f.s.

The Stroud-based collocation method uses a parallel hybrid TDIE-based field-cable-circuit simulator to

perform the deterministic EMI/EMC simulations required. S-2 and S-3 integrators require only dof 1N +

and dof2N deterministic simulations to approximate the dofN -dimensional integrals needed for computing

averages and standard deviations of pertinent observables. In practice, the proposed technique requires far

fewer deterministic simulations than Monte-Carlo or tensor-product integrators. To demonstrate the

accuracy, efficiency, and practicality of the proposed method it successfully was used to statistically

characterize coupled voltages at the feed pins of cable-interconnected and shielded PC cards as well as the

terminals of cables located in the bay of an airplane cockpit.

14

Figure Captions

Figure 1. The coupled system of equations is solved simultaneously for all field, circuit, and cable

unknowns at each time step.

Figure 2. (a) Geometry description of the RG-58 coaxial cable suspended at three nodes and the plane

wave excitation. (b) E,Re,kierr and E,Im,k

ierr , 2,3k = , 1,...,5i = , relative error between averages of the real

and imaginary parts of the coupled voltage at node 1 computed by S-2/S-3 and the tensor-product

integrators. (c) std,Re,kierr and std,Im,k

ierr , 2,3k = , 1,...,5i = , relative error between standard deviations of

the real and imaginary parts of the coupled voltage at node 1 computed by S-2/S-3 and tensor-product

integrators. (d) Error-bar plot for the average of the real part of the coupled voltage at node 1 at 100

frequencies equally located between 100 MHzf = and 600 MHzf = (computed by the S-3 integrator).

(e) Error-bar plot for the average of the imaginary part of the coupled voltage at node 1 at 100 frequencies

equally located between 100 MHzf = and 600 MHzf = (computed by the S-3 integrator).

Figure 3. (a) Geometry description of the RG-58 coaxial cable suspended at eleven nodes and the plane

wave excitation. (b) The p.d.f. of the real part of the coupled voltage at node 1 (obtained using the Monte

Carlo integrator). (c) The p.d.f. of the imaginary part of the coupled voltage at node 1 (obtained using the

Monte Carlo integrator). (d) The p.d.f. of the absolute value of the coupled voltage at node 1 (obtained

using the Monte Carlo integrator).

Figure 4. (a) Geometry description of the PC cards, RG-58 cable connecting them, and the shielding

enclosures and the plane-wave excitation. (b) Error-bar plot for the average of the real part of the coupled

voltage at node 1 at 29 frequencies equally located between 0.9 GHzf = and 2.5 GHzf = (computed

by the S-3 integrator). (c) Error-bar plot for the average of the imaginary part of the coupled voltage at

node 1 at 29 frequencies equally located between 0.9 GHzf = and 2.5 GHzf = (computed by the S-3

integrator).

Figure 5. (a) Geometry description of the cockpit, shielding boxes, RG-58 coaxial cables, and the seats

and the plane-wave excitation. (b) View from back of the cockpit: RG-58 coaxial cables (+) are located in

the partially shielded compartment under the cockpit floor; black lines represent allowed vertical

movement.

15

Figures

Figure 1

Figure 2 (a)

2 md =2 md =

φ

1 V/mEφ =

1h 2h

21

x

y

z

2R1R

k2 md =2 md =

φ

1 V/mEφ =

1h 2h

2211

x

y

z x

y

z

2R1R

k

( )EM ec EM ei e ec0 0

cte ec† CKT ch CKT CKT,nl CKT ci cte ch

te tc i CBL h te0

00

+0

l l l l

l l l l l l

l l l

ζ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− − + = + +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦

Z C I V V VC C Y C V I V I I I

C C G I V VMNA

3-D TDIE

1-D TDIE

CBL CKT coupling

Shield coupling (EM CBL)

EM CKT coupling

Shield coupling (EM CKT) →

→↔

↔

16

Figure 2 (b)

Figure 2 (c)

17

Figure 2 (d)

Figure 2 (e)

18

Figure 3 (a)

Figure 3 (b)

x

y

z

2

2R

1

1R

φ

1 V/mEφ =

k

1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md =

1h 2h 3h 4h 5h 6h 7h 8h 9h 10h

x

y

z x

y

z

22

2R

1

1R

11

1R

φ

1 V/mEφ =

k

1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md = 1 md =

1h 2h 3h 4h 5h 6h 7h 8h 9h 10h

19

Figure 3 (c)

Figure 3 (d)

20

Figure 4 (a)

Figure 4 (b)

30 cm

30 cm

18 cm 70 cm

mother boardst1 card nd2 card

1

1R

1 V/mEθ =k

θ

x

y z

2

2R

30 cm

30 cm

18 cm 70 cm

mother boardst1 card nd2 card

1

1R

11

1R

1 V/mEθ =k

θ

x

y z

2

2R

22

2R

21

Figure 4 (c)

θ

1000 V/mEθ =

RG-58 CoaxialCable Network

x

z

y

k

1.3 m

1.5 m3 m

θ

1000 V/mEθ =

RG-58 CoaxialCable Network

x

z

y

k

1.3 m

1.5 m3 m

22

Figure 5 (a)

Figure 5 (b)

1C2C 3C

4C

5C 6C

7C8C

9C

z

y

x

1C2C 3C

4C

5C 6C

7C8C

9C

z

y

x z

y

x

23

Tables

Table I. Probability density functions ( )iw x , dof1,...,i N=

( )iw x , dof1,...,i N=

Normal ( )2

21 , ,2

i i

ix

ii

e xμσ

σ π

⎛ ⎞−−⎜ ⎟⎜ ⎟⎝ ⎠ ∈ −∞ +∞

Beta ( ) ( )

( )( ) 1, ,

1, 1

i i

i i

i i i ii i i

i i i i

x a b xx a b

B b a

β α

α βα β

+ +

− −⎡ ⎤∈⎣ ⎦

+ + −

Table II. Stroud’s integration rules ( dof1,..,i N= , 1,...,k M= ) xkw i

kx ikz

S-2 S-3 S-2 S-3 S-2 S-3

Normal 1 M

i i i ik kx zσ μ= + ,

( ),ikx ∈ −∞ ∞ ,

( ),ikz ∈ −∞ ∞

Beta 1 M

( - )

2 2 3

i i i i ii kk i

a b z b ax

α

+ +=

+,

i iα β= , ,i i ikx a b⎡ ⎤∈ ⎣ ⎦ ,

[ ]1,1ikz ∈ −

( )2 1 2 12 cosr

k

r kz

Mπ− −⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )2 2 12 sinr

k

r kz

Mπ−⎛ ⎞

= ⎜ ⎟⎝ ⎠

dof1, 2, , 2r N= ⎢ ⎥⎣ ⎦… , If dofN

is odd, then ( )( )dof11 kN

kz −= −

( )2 1 2 1 22 cosr

k

r kz

Mπ− −⎛ ⎞

= ⎜ ⎟⎝ ⎠

( )2 2 1 22 sinr

k

r kz

Mπ−⎛ ⎞

= ⎜ ⎟⎝ ⎠

dof1, 2, , 2r N= ⎢ ⎥⎣ ⎦… , If dofN

is odd, then ( )( )dof11 kN

kz −= −

Table III. Values of iμ and iσ , 1,...,5i = for all five sets of simulations of the plane-wave coupling

scenario involving the RG-58 coaxial cable suspended at three nodes

Set 1 Set 2 Set 3 Set 4 Set 5 iμ iσ iμ iσ iμ iσ iμ iσ iμ iσ

1i = , 1h ( )m 0.515 0.0037 0.515 0.0075 0.515 0.015 0.515 0.03 0.515 0.06

2i = , 2h ( )m 0.515 0.0037 0.515 0.0075 0.515 0.015 0.515 0.03 0.515 0.06

3i = , 1R ( )Ω 50 0.5 50 1 50 2 50 4 50 8

4i = , 2R ( )Ω 50 0.5 50 1 50 2 50 4 50 8

5i = , φ ( )° 224 0.25 224 0.5 224 1 224 2 224 4

24

Table IV. Averages and standard deviations of the coupled voltage at the terminal of the RG-58 coaxial

cable suspended at three nodes computed by the S-2/S-3, tensor-product, and the Monte Carlo integrators

for the third simulation

superscript ( )k

( ) k3 1ReE V⎡ ⎤⎣ ⎦x

( )mV

( ) k3 1ImE V⎡ ⎤⎣ ⎦x

( )mV

( ) k3 1std Re V⎡ ⎤⎣ ⎦x

( )mV

( ) k3 1std Im V⎡ ⎤⎣ ⎦x

( )mV k M= -0.16006 0.06422 0.02460 0.01468 k P= -0.16018 0.06407 0.02435 0.01439

k S,2= -0.16019 0.06405 0.02479 0.01459 k S,3= -0.16018 0.06407 0.02433 0.01443

Table V. Averages and standard deviations of the coupled voltage at the terminal of the RG-58 coaxial

cable suspended at eleven nodes computed by the S-3 and the Monte Carlo integrators

superscript ( )k ( ) k

1ReE V⎡ ⎤⎣ ⎦x

( )mV

( ) k1ImE V⎡ ⎤

⎣ ⎦x

( )mV

( ) k1std Re V⎡ ⎤

⎣ ⎦x

( )mV

( ) k1std Im V⎡ ⎤

⎣ ⎦x

( )mV k M= 0.00591 0.00217 0.03086 0.03444 k S,3= 0.00684 0.00529 0.03131 0.03303

Table VI. Averages and standard deviations of the coupled voltage at the feed point of the PC card

computed by the S-3 and tensor-product integrators

superscript ( )k

( ) k1ReE V⎡ ⎤

⎣ ⎦x

( )mV

( ) k1ImE V⎡ ⎤

⎣ ⎦x

( )mV

( ) k1std Re V⎡ ⎤

⎣ ⎦x

( )mV

( ) k1std Im V⎡ ⎤

⎣ ⎦x

( )mV k P= 0.14671 0.23200 0.01679 0.01683

k S,3= 0.14671 0.23200 0.01033 0.01028

25

Table VII ia and ib , 1,...,9i = , for all nine random variables (The plane-wave coupling scenario

involving the RG-58 coaxial cable network located in a cockpit)

ia ib 1i = , y coordinate of the location of cable 1C ( )m 0.46875− 0.40625−

2i = , y coordinate of the location of cable 4C ( )m 0.46875− 0.40625−

3i = , y coordinate of the location of cable 5C ( )m 0.53125− 0.46875−

4i = , y coordinate of the location of cable 7C ( )m 0.53125− 0.46875−

5i = , y coordinate of the location of cable 8C ( )m 0.59375− 0.53125−

6i = , y coordinate of the location of cable 9C ( )m 0.59375− 0.53125−

7i = , y coordinate of the location of the first seat ( )m 0.342− 0.084−

8i = , y coordinate of the location of the second seat ( )m 0.342− 0.084−

9i = , arrival of excitation,θ ( )° 185 235

Table VIII. Averages and standard deviations of the absolute value of coupled voltages on the RG-58

coaxial cables situated in the bay of an airplane cockpit.

i ( )S,3iE V⎡ ⎤⎣ ⎦x ( )V ( )S,3std iV⎡ ⎤⎣ ⎦x ( )V

1 0.01581 0.02828 2 0.01150 0.02535 3 0.22904 0.31644 4 0.23250 0.31522 5 0.32650 0.14404 6 0.03390 0.14523

26

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