Homogenization Technique for Transmission Lines

with Random Permittivity Profiles

Juan S. Ochoa and Andreas C. Cangellaris Department of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign Urbana, lllinois 61801

Email: {jsochoa2, cangella} @illinois.edu

Abstract-This paper proposes a homogenization model for transmission lines with nonuniformity manifested in the random permittivity profile of their surrounding medium. The statistics of the permittivity profile are defined in terms of a finite series of correlated random variables corresponding to discrete samples taken along the longitudinal dimension of the line. Principal Component Analysis is employed to reduce the dimensionality of the random space that characterizes the uncertainty of the structure. Next, Polynomial Chaos expansion is used to capture the dependencies of the resulting homogeneous effective permit­tivity with the reduced-order random parameters that define the variability. Such construction is performed efficiently with the use of a Sparse Grid integration technique. In particular, the statistics of the propagation time of a wave traveling in the uncertain medium are calculated with the proposed homogenization model and validated with Monte Carlo simulations.

I. INTRODUCTION

Statistical analysis of transmission lines exhibiting manufac­turing variability has gained importance due to the increasing complexity and miniaturization of integrated electronics. For instance, manufacturing-induced uncertainties in material and geometric parameters affect the intended behavior of the integrating substrates, resulting in the detriment of electronic component/system performance and reduction of production yield. The predictive assessment, mitigation, and solution of such problems require appropriate models that quantify the uncertainties in terms of a finite set of random variables that can be incorporated in the electromagnetic simulation of the structure. Over the past years, several methods have been proposed to solve stochastic electromagnetic problems, including the response surface model (RSM) [1], [2], sparse grid collocation methods (SGC) [3], and polynomial chaos expansion (PC) [4], [5]. These methods aim to reduce the computational cost of traditional Monte Carlo by finding an approximate model of the desired output response over the domain defined by the input random parameters. However, despite the broad interest these methods have raised in the community, such models have not been used to assess the impact of variability that occurs within the structure itself and most applications have been limited to analyzing interconnect structures with uncertain parameters varying uniformly along the line [6]. The purpose of this paper is to present an homogenization methodology to capture the variability in the permittivity of transmission lines along their longitudinal

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dimension in terms of an effective permittivity that simplifies the statistical electromagnetic simulation of the structure.

Concerning the statistical characterization of the permittivity profile, it is presented as a correlated chain of N random variables. The dimensionality of the input space is reduced to an n dimensional random space by means of Principal Component Analysis (PCA) [7]. Therefore, the complexity of the model is reduced and subsequent calculations are simplified.

With this context in mind, we recognize that we are dealing with a non-uniform transmission line (NTL) problem. The tra­ditional approach to solve a NTL is based on the concatenation of ABCD matrices [8]. The idea of the mentioned approach is to describe the overall electromagnetic response of the system by taking the product of the individual ABCD matrices corresponding to each segment with uniform properties. In the present paper, however, we make use of a frequency­domain one-dimensional Finite Element Method (FEM) for the electromagnetic simulation in combination with a Polynomial Chaos (PC) expansion formulation [9] to characterize the impact of uncertainty on the transmission properties of the line as a function of the input random parameters. Sparse Grid integration is also employed to reduce the cost of the integration associated with the PC construction.

We find that the random non uniformity is manifested as a perturbation in the propagation constant which results in a deviation of the propagation time suffered by the wave as it travels down the random structure. The quantification of the induced jitter is necessary for the appropriate signal integrity assessment of the line since it can potentially introduce unde­sired distortion and synchronization defects.

II. FORMULATION

To demonstrate the formulation of the homogenization methodology we will employ a lossless coaxial cable example. As a starting point the variability of the permittivity is charac­terized in terms of a set of correlated random variables. Next, the dimensionality of the random space is reduced through the application of PCA which simplifies the interpolation process of the desired effective homogeneous permittivity.

A. Statistical Characterization We assume that the pennittivity of the surrounding medium

varies along the longitudinal dimension of the line (z­direction) according to the expression

(1)

where Xi is assumed to be a Gaussian random variable with zero mean and unit variance, and Erm and std(Eri) are the mean and standard deviation of the permittivity measured at every position. It is assumed that these quantities are constant for every segment of the line. The correlation between two random variables corresponding to two different positions, Zi and Zj is assumed to follow a Gaussian function that depends on the separation between such positions and is given by

'<"' . . _ e-(IZi-Zjl/C)2 L..J t] - , (2)

where parameter £ is known as correlation length and quan­tifies the size of the pennittivity fluctuations in the dielec­tric. This function ensures that the profile is continuous and smooth. Other properties, such as the conductivity, cross section or even random bending of the wires can be considered to vary in a similar fashion.

Once the pennittivity has been defined we notice that the number of random variables equals the number of elements used in the discretization of the transmission line, N. Also, the length of each discrete section, dl = LIN must satisfy the condition dl < min{AminllO, £/1O} in order to ensure that the EM solution and statistical description of the profile are accurate.

Our goal is to obtain a stochastic effective pennittivity constant that can then be used as a homogeneous property of an uniform line with equivalent transmission properties. Such permittivity is interpolated in terms of a set of basis functions of the input random variables. For a particular realization of the pennittivity profile, this homogeneous constant is obtained with an FEM solver that extracts the scattering parameters and the corresponding propagation constant from which we calculate the effective pennittivity. The computational effort required by the interpolation technique is related to the dimen­sionality of the problem. For example, in the case of a tensor grid interpolation [10], the number of required FEM simula­tions grows exponentially with the number of dimensions of the random space as qN

, where q is the number of samples taken along each random variable. Such computational barrier can be overcome by reducing the original random space of dimension N to a space of size n « N composed by the uncorrelated components presenting the largest variability. The reduced dimensionality algorithm is described next.

B. Random Space Dimensionality Reduction As already pointed out, the number of random variables

N equals the number of FEM grid segments of the structure. Since we require a relatively large number of discretized seg­ments to obtain accurate results, the dimension of the random space grows accordingly. Therefore, we take advantage of

4.5 cr-----,-----,-----.--I -.-.- x, = [·23.7, 0, 0, O[

_ x, = [0, ·20.3, 0, O[

........ x, = [0, 0, ·15.7, O[

••••••••••••••••••••••••••••• ---- Xr = [0, 0, 0, -11.1) "

, ........ -� -xr =[O,O,O,O]

���--����--��-�==������

• ;::::,:,:<':::::�:----- __ !(�L----'::'->' ...:.:;;---3.5 b....-___ c'-:-__ ..:::=:==="'=�=----::"_:_------" o 0.2 0.4 0.6 0.8

Figure 1. First 4 Principal components of permittivity profile along the transmission line.

the existing correlation between the variables to reduce the random space through a Principal Component Analysis (PCA). The purpose of PCA is to orthogonalize the random space by finding the non-correlated components of a random vector of length N, which are linear combinations of the original variables. Then, the reduced space is given by those n < N components with largest variances [7]. The reduced order random vector is found through the transformation

(3)

where random vectors X E �N, Xred E �n are related by

matrix Q E �Nxn which is built with the n eigenvectors of correlation matrix � E �NxN

of vector X. The n eigenvectors are those with largest variances, given by their corresponding eigenvalues, Ai'S. The degree of the variation captured by the reduced-order random space is given by the cumulative percentage of total variation, L:�1 Ad L:!1 Ai. This measure predicts the accuracy of the approximation and provides a criterion to decide on the dimension n of the reduced-order space. A ratio larger than 90% is recommended.

Figure 1 presents the pennittivity profiles of the first 4 prin­cipal orthogonal components of the considered transmission line of length L = 1 m, correlation length £ = 25 cm, mean pennittivity, Erm = 4 and Std(Eri) = 0.15. The ratio LI£ = 4 is the minimum number of components required to obtain an accurate model. For this example, such number of components provide a cumulative percentage variation of 94%.

C. Electromagnetic Simulation and Homogenization

With the statistical framework in hand, the electromagnetic simulation and subsequent homogenization are described. For each FEM simulation associated with the construction of the interpolation of the effective homogeneous pennittivity model, we require the position-dependent RLGC parameters of the line. In the case of a coaxial cable, there is a straightforward relation between capacitance per unit length and pennittivity,

C( ) =

27rEoEr (Z) Z In(Dld) , (4)

where D and d are the external and internal radius of the cable, respectively. When other structures are studied, for example a stripline, a polynomial expansion can be employed

to characterize the RLGC parameters in terms of the varying material properties and/or geometry as described in [4].

We proceed to find a representation of the effective homo­geneous permittivity constant in terms of a set of orthogonal multivariate polynomials. Such technique is known as Poly­nomial Chaos expansion [9] and is expressed as

Q Er(:tred) = L aiHi(-Xred)' (5)

i=O where the multivariate polynomials Hi (Xred) depend on the type of random variables. For the case of Gaussian random variables, Hermite polynomials are employed. The number of polynomials, Q, is given by (n + p)!/n!p! where p is the maximum order of the employed polynomials. At this point, we see the benefits of using PCA since it reduces the number of required polynomials. The coefficients ai's of expansion (5) are obtained by integrating in the random domain n and by making use of the orthogonality condition of the polynomials,

ai = J Er (Xred)Hi (Xred)P(Xred)dXred' (6)

n

Such integrals are efficiently computed with the use of a Sparse Grid quadrature rule, so that integral shown above is approximated to a summation of function evaluations [11]:

M

ai C::' L Er(Xted)Hi(X�ed)Wj, j=l

(7)

where weighs w/s and samples Xted'S are given by the corresponding multivariate quadrature rule.

For this type of integration scheme the number of required simulations, M is given by the number of dimensions of the random space, M cv 2knk /k! for large n, where k is known as the accuracy level of the Smolyak algorithm [12]. Again, by using PCA the number or simulations has been reduced and the model simplified.

For each iteration of our sparse grid algorithm, the EM simulator yields the scattering parameters of the structure. The (2,1)-th coefficient of the scattering matrix is obtained and used to extract the corresponding effective propagation constant of a wave traveling through the line,

The corresponding effective permittivity is computed by fitting the propagation constant to the expression

(9)

where Vm and Erm are the known propagation velocity and relative permittivity in the unperturbed line. Once we collect the M samples of Er(Xred), the coefficients (7) are evaluated and the model is complete.

III. NUMERICAL VALIDATION

In the present section we consider a numerical example to validate the proposed formulation. The inner radius of the cable is d = 0.2 cm and the outer radius, D = 0.5 cm. The capacitance per unit length varies with the permittivity (1) according to expression (4), while the inductance is constant. The length of the line is 1 m, the correlation length, 0.25 m, mean permittivity, Erm = 4 and std(Eri) = 0.15. For a maximum frequency of 4 GHz, corresponding to ,\ = 37.5 mm, we use dl = 2.5 mm which results in N = 400 segments. After the application of PCA, we obtain a 4-dimensional vector of independent Gaussian random variables with zero mean and variances Std = [12.63,10.79,8.34,5.85]. The effective permittivity is expanded in terms of 5 Hermite polynomials of the reduced-space variables Xred,

E-:" = ao + alXred,l + a2Xred,2 + a3Xred,3 + a4Xred,4, (10)

with constants ao = 3.98, al = 0.090, a2 = 0.00, a3 =

0.0156, a4 = 0.00. We notice that there is no contribution from components 2 and 4. In fact, as Fig. 1 shows, the average along the z-direction of such components is 4, meaning that the cumulative effects of components Xred,2 and Xred,4 are zero. This suggests that the permittivity can alternatively be estimated with the mean value of the randomly generated profile of the line; therefore, avoiding the electromagnetic simulation. By using the mean of the permittivity of the segments of the line to estimate the effective permittivity for a specific realization of the profile instead of the electromagnetic simulation (8), the coefficients of model (10) are ao = 4, al = 0.091, a2 = 0.00, a3 = 0.0159, a4 = 0.00. There­fore, the mean-based approach provides with an accurate and significantly more expeditious way to calculate the effective permittivity.

The impact of the variability in the permittivity of the line is quantified in terms of perturbation in the expected propagation time as observed in Fig. 2, where a number of Monte Carlo simulations of the far-end time-domain voltage are pictured for coaxial cables with random permittivity profiles. For each of those iterations, we perform an FEM simulation to obtain the scattering parameters of the entire line with a randomly generated profile which are used to find the time-domain responses for a given load conditions and source. The cable is driven by a voltage source generating a rectangular pulse of amplitude 1 V and tum-on delay time of 1 ns, rise and fall times of 2.5 ns, and width of 3 ns. The source impedance is 50n and the termination impedance is also 50n.

The results obtained with the Monte Carlo simulation are used as a reference to validate our model. By employing the homogeneous model, the time deviation is computed as follows:

0.5 Unperturbed Line

0.4 ,,� 0.3 c: CU Monte Carlo CUe> Simulations-�

mEl 0.2 u.o

>

0.1

0 0 2 4 6 10 12 14

Figure 2. Far end voltage samples for a number of Monte Carlo realizations of a lossless coaxial cable. The distortion in the line is manifested in terms of a random deviation of the time delay, flt.

4

o � t(ns)

-- Monte Carlo EM simutation­based model Mean-based model

L-� ________ � ________ -L ________ � ________ L-__ � -0.2 -0.1 o 0.1 0.2

Figure 3. Probability density function of the deviation of the time delay induced due to the variability in the permittivity of the line. The methodology based on stochastic collocation is compared with Monte Carlo simulation.

The probability density function of tlt as a result of the Monte Carlo simulation is compared with the corresponding density obtained with our homogeneous stochastic model and the curves are shown in Fig. 3. Very good agreement is observed. The advantage of employing the proposed method­ology is clear if we consider the computational savings. For the traditional Monte Carlo method, 10,000 FEM simulations where employed to construct the corresponding PDF which

took 6.05 hours to implement in a MATLAB® code running on a 1.80 GHz Xeon CPU Windows machine, while only 33 simulations corresponding to 2.74 minutes were needed to construct the polynomial chaos interpolation of the effective relative permittivity constant with Smolyak accuracy level of 3 for the electromagnetic-based model. On the other hand, by using the mean-based approach, the time is cut down to 6 x 10-3 seconds.

IV. CONCLUSIONS

In this paper we proposed a methodology to account for the random permittivity profiles in transmission lines and the impact on their transmission properties. As a result, an effective homogeneous permittivity model is constructed in terms of orthogonal polynomials. The construction of the model has been expedited by using a dimensionality reduction algorithm and an efficient multivariate integration technique based on the Smolyak algorithm, which reduces the number of required simulations to achieve a certain accuracy. The homogeneous model can be used to estimate the impact on the propagation time of a wave when it travels through the

disordered structure, which might result in the distortion of the signal and desynchronization of the system.

In the proposed model we have neglected the impact of mul­tiple reflections encountered by a wave as it travels through the nonuniform line. These reflections can be quantified in terms of an effective attenuation constant resulting in a complex effective permittivity. The lack of this effective attenuation, confirmed by our electromagnetic simulations depicted in Fig. 2, is attributed to the smoothness of the particular profile we have employed. For other cases, however, where changes are abrupt, the characterization of the reflections becomes imperative. Such models are currently under consideration and will be presented in a future publication.

ACKNOWLEDGMENT

This material is based upon work supported by, or in part by, the US Army Research Laboratory and the US Army Research Office under grant number W911NF-IO-1-0269.

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