# Methods for Controlling Electrostatic Discharge and Electromagnetic Interference in Materials

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## Abstract

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## 1. Introduction

## 2. Electrostatic Discharge and EM/RF Interference in Materials

## 3. Effective Medium Theory

#### 3.1. Effective Medium Theory of Inhomogeneous Hyperspheres

#### 3.2. The Limit ${\lambda}_{21}\to \infty $: Maximising the Dielectric Function

#### 3.3. The Limit ${\lambda}_{11}\to 1$: The Perfect Conducting Limit

#### 3.4. The Limit ${\lambda}_{31}\to 0$: EMI/RFI Cancellation

## 4. Transformation Medium Theory

#### 4.1. Determining the Equations and Geodesics of Electromagnetic Fields in a Material

#### 4.2. Transformation of the Constitutive Parameters in Maxwell’s Equations

#### 4.3. Controlling EM Fields in Different Regions of a Material

#### 4.4. Propagation of the Fields along Their Geodesics in Inhomogeneous Materials

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The constitutive parameters of an inhomogeneous (two-layer) hyperspherical inclusion. A homogeneous hypersphere corresponds to the case where ${a}_{1}={a}_{2}$, hence ${\u03f5}_{1}={\u03f5}_{2}$ and ${\mu}_{1}={\mu}_{2}$. The surrounding material has parameters ${\u03f5}_{0}$ and ${\mu}_{0}$.

**Figure 2.**The dielectric function when ${\u03f5}_{1}$ is given by (45) as a function of ${\u03f5}_{2}$ and hyperinclusion dimension. Here, the offset from the singularity is taken to be $\delta \u03f5=0.001$ and $\beta =0.9$. (

**a**): the variation in the dielectric constant for some hyperspherical inclusion dimensions; (

**b**): the relation between the constitutive parameters ${\u03f5}_{1}$ and ${\u03f5}_{2}$ for a spherical inclusion.

**Figure 3.**(

**a**) Elimination of the electromagnetic field in a inhomogeneous spherical inclusion ($d=3$) with a thin outer region $\beta \to 1$ in a material with ${\u03f5}_{0}=1$ at a frequency of $f=0.8$ GHz. (

**b**) A closer look at the same inclusion when the outer region is thicker, $\beta \to 0$. In both cases, the field inside the inclusions has been cancelled and is zero, while the field outside has a maximum value.

**Figure 4.**Homogeneous $d=2$ perfect conducting inclusions are shown, i.e., in the limit $\beta \to 1$. The electric field lines (not shown) are perpendicular to the potential surfaces caused by a positive charge. (

**a**) No inclusions, (

**b**) 2 inclusions, (

**c**) 4 inclusions, (

**d**) 10 inclusions.

**Figure 5.**Using 30 perfect conducting inclusions, both the potential and electric field are eliminated in the oval region shown.

**Figure 6.**The variation in the hyperspherical inclusion constitutive parameters ${\u03f5}_{1}$ and ${\u03f5}_{2}$ for different dimensions that cancel EM/RF interference. Notice that both $({\u03f5}_{1},{\u03f5}_{2})$ are always positive.

**Figure 7.**(

**a**) Scattering of a homogeneous spherical inclusion ($d=3$) by a transient EM field at a frequency of $f=2$ GHz for $a=5$ cm and ${\u03f5}_{1}=5$, while the material it is embedded in has a permittivity value of ${\u03f5}_{0}=1$ (vacuum or air). (

**b**) Scattering of the same inclusion when it is inhomogeneous for ${a}_{1}=2.5$ cm, ${a}_{2}=5$ cm, ${\u03f5}_{1}=-6.163$, and ${\u03f5}_{2}=5$, respectively.

**Figure 8.**(

**a**) Scattering of a transient electromagnetic field of frequency $f=2$ GHz travelling from the left by homogeneous $d=3$ inclusions in an array inside a material medium with ${\u03f5}_{0}=2.5$. The inclusions have a radius of ${a}_{2}=a=5$ cm and ${\u03f5}_{1}=5.0$. (

**b**) Elimination of the forward scattered EMI/RFI to the right by the inclusion array where ${a}_{1}=4.409$ cm, ${a}_{2}=5$ cm, ${\u03f5}_{1}=2$, and ${\u03f5}_{2}=5$ for each of the inclusions in the array. Note that all the constitutive parameters are positive.

**Figure 9.**(

**a**) A homogeneous material can be transformed so that it exhibits compression. (

**b**) The material properties have been changed by the transformation of the permittivity (or permeability) according to (117).

**Figure 10.**(

**a**) A homogeneous material is represented by orthogonal Cartesian coordinates. (

**b**) The material properties have been changed by the transformation of the permittivity (or permeability) tensor by a coordinate transformation as given by (118). With parameters $a=1$ and $b=1$, the material now behaves differently compared to the homogeneous case. The material has undergone expansion.

**Figure 11.**The constitutive tensors of an inhomogeneous material ${\u03f5}^{{i}^{\prime}{j}^{\prime}}={\mu}^{{i}^{\prime}{j}^{\prime}}$ that undergoes expansion; see (131).

**Figure 12.**The constitutive tensors of an inhomogeneous material under compression via ${\u03f5}^{{i}^{\prime}{j}^{\prime}}={\mu}^{{i}^{\prime}{j}^{\prime}}$ under the transformation coordinates (117). The tensor components ${\u03f5}^{{x}^{\prime}{x}^{\prime}}$ and ${\u03f5}^{{z}^{\prime}{z}^{\prime}}$ show a localised variation in the permittivity approaching infinity just as in the case of inclusions in a medium as discussed in Section 3. Here, ${\sigma}_{x}={\sigma}_{y}=0.5$.

**Figure 13.**(

**a**) An electromagnetic wave propagating in a homogeneous medium at a frequency of $f=0.1$ GHz. (

**b**) The same electromagnetic wave propagating inside the modified medium which has undergone an expansion of its geometry via a transformation of the permittivity (or permeability) according to (118). Notice that in this case, the field is reduced in the middle regions of the medium close to or equal to zero. Here, $a=2$ and $b=1/2$.

**Figure 14.**(

**a**) An electromagnetic wave propagating in a homogeneous medium at a frequency of $f=0.1$ GHz. (

**b**) The same electromagnetic wave propagating inside the modified medium which has undergone a compression of its geometry via a transformation of the permittivity (or permeability) according to (117) with ${\sigma}_{x}={\sigma}_{y}=0.5$.

**Figure 15.**(

**a**) An electromagnetic wave propagating in a homogeneous medium at a frequency of $f=0.1$ GHz. (

**b**) The same electromagnetic wave propagating inside the modified medium which has undergone a transformation of the permittivity (or permeability) according to (117) where ${\sigma}_{x}={\sigma}_{y}=5$.

**Figure 16.**(

**a**) An electromagnetic wave propagating in a homogeneous medium at a frequency of $f=0.1$ GHz. (

**b**) The same electromagnetic wave propagating inside the modified medium which has undergone an expansion of its geometry via a transformation of the permittivity (or permeability) according to (139).

**Figure 17.**(

**a**) A homogeneous material can be transformed so that it exhibits compression and expansion, a kind of warping effect. (

**b**) The material properties have been changed by the transformation of the permittivity (or permeability) according to (145).

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**MDPI and ACS Style**

Alexopoulos, A.; Neudegg, D.
Methods for Controlling Electrostatic Discharge and Electromagnetic Interference in Materials. *Foundations* **2024**, *4*, 376-410.
https://doi.org/10.3390/foundations4030025

**AMA Style**

Alexopoulos A, Neudegg D.
Methods for Controlling Electrostatic Discharge and Electromagnetic Interference in Materials. *Foundations*. 2024; 4(3):376-410.
https://doi.org/10.3390/foundations4030025

**Chicago/Turabian Style**

Alexopoulos, Aris, and David Neudegg.
2024. "Methods for Controlling Electrostatic Discharge and Electromagnetic Interference in Materials" *Foundations* 4, no. 3: 376-410.
https://doi.org/10.3390/foundations4030025