Multiscale electromagnetic modelling of metal braids

H. Schippers1 J. Verpoorte2

1 National Aerospace Laboratory NLR, Avionics Division, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands, e-mail: Harmen.Schippers@nlr.nl, tel.: +31 88 511 4635, fax: +31 88 511 4210. 2 National Aerospace Laboratory NLR, Avionics Division, Anthony Fokkerweg 2, 1059 CM Amsterdam, The Netherlands, e-mail: Jaco.Verpoorte@nlr.nl, tel.: +31 88 511 4460, fax: +31 88 511 4210.

Abstract − This paper describes the modelling to calculate the induced currents and voltages inside avionics boxes of aircraft due to high intensity radiated electromagnetic fields. The modelling involves several length scales: the wavelength of the exterior electromagnetic field, the diameter of the tubular braid, and the diameter of the wires in the carriers of the braid. The modelling approach is used to calculate the induced voltage at the interfaces of two avionics boxes which are connected to each other by a cable protected by a metal braid.

1 INTRODUCTION

High intensity electromagnetic fields outside aircraft can penetrate through windows and other apertures inside cabin and cockpit. The metal housing of avionics boxes and the metal braids around cables should provide electromagnetic shielding for a range of RF frequencies. The shielding effectiveness of metal braids of cables is governed by the geometry and the materials of the braid. The shielding effectiveness can be characterized by the transfer impedance of this metal braid. The transfer impedance can be calculated for a range of frequencies by appropriate analytical models or advanced numerical finite element models. The diameter of wires of typical braids is about 0.2 mm. These braids contain small rhombic apertures with a width similar to the diameter of the wires. The transfer impedance is used in the HIRF certification process to determine the common mode voltage in the cable due to the current flowing through the shield.

X-Axis

Y-Axis Z-Axis

Figure 1 Test case: two metallic boxes connected by

a cable with metal braid (Courtesy of EMCC)

This paper will address the calculation of induced voltages inside two avionics boxes which are

connected by a cable (length 1 m) protected by a metal braid (see Figure 1).

2 MULTIPLE SCALES

The mathematical modelling of penetration of exterior electromagnetic fields into the interior of the braid contains several length scales: the wavelength of the exterior electromagnetic field, the diameter of windows in aircraft, the diameter of the tubular braid, and the diameter of the wires in the carriers of the braid. The propagation of electromagnetic fields into the aircraft is governed by the Maxwell equations in the frequency domain

2E k E∇×∇× = (1) with 2 /k π λ= , and λ the wavelength of the exterior electromagnetic field. For radio frequencies of interest the wavelength can be between 10 cm and 100 meter. In case the wavelength is of the same order of magnitude as the apertures in the fuselage of the aircraft, the exterior field can generate fields inside the aircraft with rather high amplitudes. An example of propagation of waves through a window is presented in Figure 2. This figure shows a cross section of a fuselage with a metallic floor. The results were obtained by 2D calculations. Hotspots with high amplitude can be observed from Figure 2. The amplitudes of the fields are relevant for the design and development of shields for avionics, cables (wiring) and connectors. The shielding effectiveness of cables depends on the geometry and materials of metal braids. A close-up of a typical metal braid is shown in Figure 3. The diameter of the braid can be between 8 mm and several centimetres. The diameter of wires of typical braids is about 0.2 mm. These braids contain small rhombic apertures with a width similar to the diameter of the wires.

978-1-61284-978-2/11/$26.00 ©2011 IEEE

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Figure 2 Calculation of propagation of

electromagnetic field through window of fuselage

Figure 3 Close-up of metal braid

3 MODELING

For the calculation of fields, incident on avionics boxes and metal braided cables, many numerical techniques can be used, such as classical finite element methods, boundary element methods, asymptotic ray-tracing methods and hybrid techniques. The applicability of these techniques depends on the frequency of the exterior High Intensity Radiated Field (HIRF), the dimensions of the aircraft and the availability of computational recourses. In this paper a boundary element method is used to calculate the incident electric field on the metal braid of a cable between the two boxes of the test case as shown in Figure 1. Surface grids on the boxes are generated by means of the GiD tool of CIMNE (Ref. [1]). GiD is a universal, adaptive and user-friendly pre and postprocessor for numerical simulations in science and engineering. Between these boxes the cable braid has been approximated by wire elements. The surface grids on the boxes are displayed in Figure 4. The induced currents on the surface elements and wire segments are computed by the CONCEPT-II code of University of Harburg (see Ref. [2]). The CONCEPT-II code is based on the method of moments in the frequency domain. Integral equations are solved numerically for surface and wire currents. For a plane wave field with amplitude 1 kV/m and a frequency 1 GHz the induced wire currents are shown in Figure 5.

Figure 4 Surface grids on boxes of Figure 1

Figure 5 Induced currents on metal braid of cable between on boxes of Figure 1

The induced currents on the cable braid and the transfer impedance of the metal braid determine the induced voltage of the cables inside the braid via

/ tV x IZ∂ ∂ = , with tZ the transfer impedance. Analytical formulations (see Ref. [9]) are used to calculate the transfer impedance of the braid. Because the cables inside the braid are connected to components inside the boxes, the current in these cables will generate an electric field inside the boxes, which can once again be calculated by numerical techniques.

A metal braid is completely described by 6 parameters (see Figure 6 and Figure 7). These parameters are:

• Diameter D of braid (real number, dimension meters)

• Number of carriers C (i.e. belts of wires) in the braid (integer number)

• Number of wires N in a carrier (integer number) • Diameter d of a single wire (real number,

dimension meters) • Conductivity σ of the wires (real number,

dimension S/m)

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• Weave angle α of the braid (real number, degrees)

Figure 6 Metal braid for cable (diameter of braid is D)

Figure 7 Characteristics of metal braid

Analytical models for calculation of the transfer impedance contain in general two components, one part ( dZ ) representing diffusion of electromagnetic energy through the metal braid, and a second part ( j Mω ) representing leakage of magnetic fields through the braid

t dZ Z j Mωυ= + (2) with υ the number of holes per unit length. The diffusion component dZ of the metal braid is governed by the DC resistance of the metal braid and diffusion of waves through the wall of the cylindrical braid. Following Ref [3] the diffusion can be computed by

0 sinhd

dZ Rd

γγ

= (3)

where d is the thickness of the wires in the metal braid, and γ is the complex propagation constant of the wires ( (1 ) /jγ δ= + , with δ the skin depth of

the wire, 2 /δ ωμσ= ). The resistance 0R is

computed per unit length. The DC resistance 0R is governed by the conductivity σ and the averaged

cross section of the braid. With reference to [3] the resistance per unit length reads

0 2

4cos( )

Rd NCπ σ α

= (4)

The second term of (2) is governed by the

inductance of magnetic fields through the apertures in the metal braid. The inductance is a local phenomenon. Expressions for the inductance can be derived by considering the inductance through a single aperture and then superimposing the contributions of all apertures. Hence, the interaction of induced magnetic fields through neighbouring apertures is usually neglected. In general, the inductance has two parts: hole inductance and braid inductance. The hole inductance hM is caused by penetration of magnetic fields through the rhombic apertures in the metal braid (see Figure 7), while the braid inductance bM arises from the woven nature of the braid. In the semi-empirical models as described by [5], a third inductance term is introduced, the skin inductance sM , which is due to eddy current in the walls of the rhombic apertures. In summary, the inductance M in equation (2) is the superposition of the hole inductance hM , braid inductance bM and

skin inductance sM ,

h b sM M M M= + + (5) The hole inductance follows from the propagation

of magnetic fields through small aperture of braids. Consider a coaxial cable system with centre wire and cylindrical shield with zero thickness and radius

/ 2a D= and containing a circular hole with radius

0r (see Figure 8). In the vicinity of the aperture, the

magnetic field can be written as H X= −∇ . In the domain around the aperture, the potential X satisfies Laplace equation. Along the closed surface of the

braid /X n∂ ∂ vanishes since H is tangential there. In the exterior space, far away from the aperture, the potential X has to satisfy (see Ref. [8])

0lim sin sinr

X H r θ ϕ→∞

= (6)

Figure 8 Coaxial cable (radius a ) with induced magnetic field

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In the interior space, the potential X has to disappear. This problem was solved analytically by Kaden in Ref. [8].

The hole inductance is determined by calculating the voltage that is induced by the magnetic field 0H due

to a current I in the cylindrical wall given by

0IHDπ

= (7)

The induced voltage V in the wire of the cable system can be determined using the analytical solution of potential X . By Ref. [8] the potential V in the wire centre is

0 0

2j mHV

Dω μ

π= (8)

where m is the magnetic polarizability of the circular hole given by 3

08 / 3m r= . The inductance for a circular hole follows from substituting (7) and (8) into /M V j Iω= . The result is

3

0 0 02 2 2 2

42 3

m rMD D

μ μπ π

= = (9)

Equation (9) shows that the hole inductance depends linearly on the magnetic polarizability. The effect of the wall thickness of the braid can be taken into account by multiplying the right-hand side of (9) with an attenuation factor exp( )τ− due the so-called “chimney” effect (see Ref. [8]).

104

105

106

107

108

10-3

10-2

10-1

100

Frequency (Hz)

Zt

(ohm

/m)

Sample nr.8, beta:1

BEATRICS

ERMESMeasurements

Figure 9 Calculated and measured transfer impedance

of typical metal braid

Analytical expressions for the magnetic polarizability of elliptical apertures have been presented in Ref. [3] and [4]. Numerical tools for the calculation of the magnetic polarizability for small apertures with arbitrary shape have been presented in Ref [6] and [7]. In these papers dimensionless magnetic polarizabilities of apertures were introduced by

introducing the length scale S with S the surface of the aperture. It was shown in Ref [7] that the

dimensionless magnetic polarizabilities of elliptical and rhombic apertures are almost equal. More details about the calculation of hole and braid inductance have been presented in Ref. [9]. Some results of calculations and measurements of transfer impedance of a typical metal braid are displayed in Figure 9.

4 Conclusion

The multi-scale problem of modelling the induced voltage in a shielded cable, due to an external electromagnetic field, has been described. The modelling of the transfer impedance of the shield was described in detail in section 3. The induced voltage between the internal wires and the braid can now be calculated via / tV x IZ∂ ∂ = . The shield current I can be computed as explained in section 2 of this paper. Once the voltage at the interface between the cable and the avionics box is know, the currents and electromagnetic fields inside the boxes can be determined.

Acknowledgment

The work described in this paper and the research leading to these results has received funding from the European Community's Seventh Framework Programme FP7/2007-2013, under grant agreement no. 205294, HIRF SE project.

References

[1] GiD Pre- and post processing tool, see http://gid.cimne.upc.es/ [2] CONCEPT II, see www.tet.tu-harburg.de/concept [3] Vance, E.F., Shielding effectiveness of braided-wire shields,

Interaction Note 172, Stanford Research Institute, April 1974. [4] Vance, E.F., Shielding effectiveness of braided-wire shields,

IEEE Transactions on, Electromagnetic Compatibility, Vol.17, no. 2, May 1975, 71-77

[5] Kley, T., Optimized Single-Braided Cable Shields, IEEE Trans. On EMC, Vol. 35, No. 1, February , 1-9, , 1993

[6] De Smedt, R.; Van Bladel, J.; Magnetic polarizability of some small apertures, IEEE Transactions on Antennas and Propagation, Issue Date: Sep 1980 Volume: 28 Issue:5, page(s): 703 - 707

[7] F. De Meulenaere and J. Van Bladel; Polarizability of some small apertures, IEEE Trans. Antennas Propagat., vol. AP-25, pp. 198, 1977.

[8] H. Kaden, Wirbelströme und Schirmung in der Nachrichtentechnik, 1959, new edition 2006, Springer Verlag

[9] H. Schippers, J. Verpoorte, R. Otin; Electromagnetic Analysis of Metal Braids, Paper to be presented at IEEE EMC Symposium, York, UK, 2011.

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