IEEE ELECTRICAL DESIGN OF ADVANCED PACKAGING AND SYSTEMS (EDAPS) SYMPOSIUM 1

Mapping the Foam-induced Dielectric Anisotropyfor High-Speed Cables

Qiwei Zhan∗, Rui Zhang†, James Baker†, Hansen Henning†, and Qing Huo Liu∗∗Department of Electrical & Computer Engineering, Duke University, Durham, NC 27708, USA

†Lorom Industrial Co. Ltd., Morrisville, NC 27560, USA

Abstract—Foaming the expanded polyethylene (PE) is a pre-vailing way to lower the dielectric constants and dissipationfactors of insulators in modern communication high-speed cables.The porosity and elongated cell shapes impose significant effectson the dielectric properties of PE. However, a theoretical studyis lacking. This work proposes an effective way to predict thedielectric parameters of foamed PE. Unlike directly generatingthe extremely dense meshes for the random distributed pores,which is usually computationally intractable, we apply theelectrodynamics homogenization method to obtain the equivalentanisotropic model for the complex foamed insulators. It is shownthat the predicted values are amenable to the experimentalmeasured data.

Index Terms—high-speed cables; extrusion foaming process;electrodynamic homogenization; anisotropic dielectric media.

I. INTRODUCTION

High-speed cables are widely used to transmit electronicsignals in various environment, such as data center cabling,medical instruments, and automotive products. Two criteriaexist for an excellent cable: a low dissipation factor and a lowdielectric constant. These goals can be achieved by foamingthe expanded polyethylene (PE), which insulate the innercoppers from the outside shielding metals. Parenthetically, thefoamed insulations allow the weight reduction and volumeminiaturization of cables.

Due to the extrusion foaming process over the expanded PE,the cells are elongated preferably along the axial direction. Inaddition, even with random distribution functions, the poresalso have significant effects on the dielectric properties.

For the foamed electrical lines, various research has beeninvestigated. [1] performs experimental study and gives themeasured dielectric data. [2] models the exact foam insulatingmaterial for the gas insulated transmission lines with the finiteelement method.

However, a theoretical prediction of the foamed insulationis still lacking. In this research, we propose an effective wayresorting to the homogenization method, where an equivalenthomogeneous medium is used to approximate the complexporous composites. One basic assumption for homogenizationis that the wavelength must be much larger the scale ofinhomogeneity for the composite materials.

The electromagnetic homogenization is a venerable realm,which is even discussed before the formalism of Maxwell’sequations [3, p. x]. In retrospect, remarkable milestones ofthe development of homogenization include self-consistentapproximation (SCA) method (i.e., Bruggeman method) and

differential effective medium (DEM) method (i.e., Maxwell-Garnett method, average t-matrix approximation) [3], [4], [5],[6], [7].

Different homogenization methods are all contingent uponthe calculation of the depolarization dyadics (or Eshelby ten-sor). For low-frequency or electrostatic problems, [6], [7] pro-pose compact formulations, which include the depolarizationfactor, the depolarization dyadics, and polarization dyadics.Towards higher frequencies, [3] summarizes their researchachievement of mixing formulas in the past two decades. Inmost general, the inclusion in the composites can be arbitrarilyshaped; whereas the most complex but analytically extricablegeometry is the ellipsoid. Due to the preferably orienteddistribution of the small particles, the anisotropic effects areintroduced. However, in the case of random distribution orsphere-shaped inclusions, the composites remain isotropic butthe properties are neither one phase of the composites.

This paper is organized as follows. In Section II, we providea compact form for the two homogenization methods. A de-tailed expressions are provide in Section III. Section IV showsnumerical validations with experimental data. Furthermore, weprovide the anisotropic simulation in Section V. Conclusionremarks are drawn in Section VI.

II. HOMOGENIZATION FORMALISMS

A. Self consistent approximation (SCA)

The SCA treats the two phases of composites equally.It does not need to set the background, but the shape andorientation of every component are required.

Ceff = [(1− φ)C1Q1 + φC2Q2][(1− φ)Q1 + φQ2]−1 (1)

where C is the third-order dielectric/magnetic tensor or sixth-order bianisotropic tensor, φ is the porosity, and the Eshelby-related term Q depends on Ceff in the previous iteration, thedetailed expression will be explored in Section III.

B. Differential effective medium

The DEM depends on the inclusion adding orders: differentorders will lead to different results.

d CDEM

d v=

1

1− v(Ci −CDEM) Qi (2)

where v is the concentration of the inclusion. This ODE canbe solved numerically, such as applying Runge-Kutta methods[8].

IEEE ELECTRICAL DESIGN OF ADVANCED PACKAGING AND SYSTEMS (EDAPS) SYMPOSIUM 2

x y

z

oUx Uy

Uz

Fig. 1. An ellipsoid with its principle axes along the Cartesian x, y, z axes.

III. ESHELBY-RELATED TENSORS

For an ellipsoidal inclusion shown in Fig. 1, the surface isparameterized by

re(θ, φ) = ρUr(θ, φ) (3)

with

U =1

3√UxUyUz

UxUy

Uz

(4)

and r(θ, φ) is the spherical unit vector with the radius ofρ. The polarizability density is

(Ci −CDEM

)Qi, where the

Eshelby-related term reads

Qi =[I + P (Ci −CDEM )

]−1(5)

withP = iωD (6)

where the depolarization dyadics reads

D(ρ) =D0 +D+(ρ) (7)

Here we only consider the anisotropic dielectric composites.The first term in (7) is associated with the vanishingly smallinclusions, having the expression as

D0 =1

4πiω

2πˆ

0

dφ

π

0

dθ sin θ

(1

tr(ε ·A)A

)(8)

where

A =

sin2 θ cos2 φ

U2x

sin2 θ sin2 φU2

y

cos2 θU2

z

(9)

For the second term related to the finite volume in (7), withthe application of residue calculus, we refer to [9].

The above depolarization dyadics expression is based onelectrodynamics, for the simpler electrostatics-based formula-tion, we refer to [6], [7].

TABLE IMPE AT 34 KHZ AND DIFFERENT TEMPERATURES 60 AND 80◦C

Porosity(%) 0 10 20 28ε′ 2.32 2.25 2.00 1.83ε′′60◦C × 106 531 487 — 314ε′′80◦C × 106 624 599 481 350

(a)

(b)

(c)

Fig. 2. Comparisons of measured data and predicted data: (a) real part, (b)imaginary part at temperature 60◦C, (c) imaginary part at temperature 80◦Cof MPE, when the operating frequency is 34 kHz.

TABLE IIMPE AT THE OPERATING FREQUENCY 11.2 GHZ

Porosity(%) 0 10 28ε′ 2.20 2.02 1.79ε′′ × 106 600 520 370

IEEE ELECTRICAL DESIGN OF ADVANCED PACKAGING AND SYSTEMS (EDAPS) SYMPOSIUM 3

(a)

(b)

Fig. 3. Comparisons of measured data and predicted data: (a) real part, (b)imaginary part of MPE at the operating frequency 11.2 GHz.

IV. VALIDATIONS OF THE MIXTURES

From the optical micrographs shown in [1, Fig. 2], weestimate the ellipsoidal inclusion as Ux = Uy = Uz/3.From the measured data provided in [1], we observe thatthe dielectric constants depend on the operating frequency.In addition, the operating temperature also shows significanteffects on the imaginary part of permittivity. Table I and IIprovide the real imaginary parts of the dielectric constants.Fig. 2 and 3 show the comparisons between the predictedvalues and the experimental measured data: good agreementsare achieve, given the measured dielectric values are the trans-verse components, rather than the axial component. Especially,for high operating frequency (i.e., 11.2 GHz), the misfits arequite small. In addition, the Wiener bounds are plotted to givethe regularization ranges [7, p. 152]. Even the formulationsprovided in Section III do not have an explicit indication theeffects, at a specific frequency, we can measure the dielectricproperties with a specific porosity, then predict the permittivityat any porosities.

V. ANISOTROPIC SIMULATION

In this research, we simulate the twin-ax cable with HFSS.The insulator is anisotropic with the dielectric constant asdiag(2.1+ 0.001j, 2.1+ 0.001j, 3.1+ 0.004j). The excitationare two differential pair wave ports. The operating frequency is

Fig. 4. Electrical field magnitude distribution.

100 GHz, along the axis direction, about 5 wave-length profile(10 mm) is observed.

VI. CONCLUSIONS AND PERSPECTIVES

In this research, we apply a homogenization theory to pre-dict the porous dielectric media. The theoretical values matchthe experimental data very well, which are rarely touched byeither industrial companies or academic institutes.

REFERENCES

[1] M. Straat, I. Chmutin, and A. Boldizar, “Dielectric properties of polyethy-lene foams at medium and high frequencies,” Annual transactions of thenordic rheology society, no. 18, pp. 107–117, 2010.

[2] H. Pendse and G. Karady, “Development of the insulating foam model forthe study of electric field distribution for the application of gas insulatedtransmission line,” in Power & Energy Society General Meeting. IEEE,2010, Conference Proceedings, pp. 1–5.

[3] T. G. Mackay and A. Lakhtakia, Modern analytical electromagnetichomogenization. Morgan & Claypool Publishers, 2015.

[4] B. E. Hornby, L. M. Schwartz, and J. A. Hudson, “Anisotropic effective-medium modeling of the elastic properties of shales,” Geophysics, vol. 59,no. 10, pp. 1570–1583, 1994.

[5] G. Mavko, T. Mukerji, and J. Dvorkin, The rock physics handbook: Toolsfor seismic analysis of porous media. Cambridge University Press, 2009.

[6] O. Durr, W. Dieterich, P. Maass, and A. Nitzan, “Effective medium theoryof conduction in stretched polymer electrolytes,” The Journal of PhysicalChemistry B, vol. 106, no. 24, pp. 6149–6155, 2002.

[7] A. H. Sihvola, Electromagnetic Mixing Formulas and Applications. TheInstitution of Electrical Engineers, 1999.

[8] Y. Q. Zeng and Q. H. Liu, “A multidomain PSTD method for 3D elasticwave equations,” Bull. Seismol. Soc. Am., vol. 94, no. 3, pp. 1002–1015,2004.

[9] T. G. Mackay, “Depolarization volume and correlation length in thehomogenization of anisotropic dielectric composites,” Waves in randommedia, vol. 14, no. 4, pp. 485–498, 2004.