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## Analytical methods: the diffraction theory

Several models can describe diffraction phenomena, their choice depends on the size of the aperture. At first, general purpose principles will be exposed, then specific models suitable for small electric dimensions will be reminded.

### Description of general apertures

The description of the electromagnetic field behind an aperture is a complex procedure [Bouwkamp, 1954]. Small apertures can be described from the Bethe’s theory (which will be reminded in the next section), while large ones may be considered by means of the geometrical theory of diffraction [Balanis, 2016, chap. 12.10]. For aperture dimensions close to the wavelength of interest, the two following principles apply.

The Huygens equivalence principle [Balanis, 2016, chap. 12.1] Assume that electric ~ J1 and magnetic ~M1 sources (as current densities) generate electric ~E1 and magnetic ~H1 fields everywhere.

Consider the situation depicted in Fig. I.2.1 which represents a closed volume ¯ containing the electric ~ J1 and magnetic ~M1 current densities. The volume outside ¯ is referred as . An equivalent configuration that generates ~E1 and ~H1 outside ¯ consists in removing the ~ J1 and ~M1 sources from ¯ , and to impress magnetic ~M and electric ~ J current densities at the boundary S between the two volumes.

The new fields inside ¯ are ~E and ~H . ~M and ~ J are expressed from the tangential fields to ¯ as: ~ J = ~n ∧ (~H1 − ~H ) (I.2.1) ~M −~n ∧ (~E1 − ~E ).

#### Equivalent electric and magnetic dipoles for small apertures

Some simple geometry apertures, may be modeled by electric and magnetic dipoles. This method was introduced in [Bethe, 1944] but is limited to apertures with small electrical dimensions: the condition is that the maximum transversal dimension dmax of the aperture should be smaller than λmin 2π , with λmin the shortest wavelength of interest. If so, the aperture may be replaced by an elementary electric dipole of moment ~Pe and an elementary magnetic dipole of moment ~Pm while shorting the aperture with a metallic wall. The expressions of these elementary moments are: ~Pe = ǫαe~Esc (I.2.8) ~Pm = −α ~H sc.

#### Topological formalism associated to the power balance approach

The power balance (PWB) approach was set by Hill (see [Hill, 2009, chap. 8.2]) with the intent to qualify energy transfers inside reverberant environments at a macroscopic scale. Thus, instead of determining electric/magnetic fields or voltage/current values, electromagnetic environments are identified by means of average power values. It models transfer of energy into cavities from a macroscopic viewpoint. To be valid, the model requires that the environment is overmoded such that the field inside the reverberant environment is uniformly distributed in polarization and in phase. It is thus mandatory that the size of the cavity is large compared to the longest studied wavelength. Applying this method involves to solve an equilibrium equation that describes energy transfers inside a cavity. The PWB states that the sum of the energy losses and energy transfers in a cavity must be equal to the injected power Pt: Pt = X iPdi.

**Applications of the PWB approach associated to the BLT formalism**

This method was successfully employed in [Junqua et al., 2005] and [Junqua, 2010] to determine the mean power in several environments, such as building rooms, cavities in missiles, or avionic equipment casings. In every scenario, the mean power inside the N cavities of the system was determined with a good agreement up to 12GHz. The resolution of this multi-cavity problem consists in solving a system of N linear equations and provides the mean power density in each cavity. It is possible to solve this problem by taking advantage of the software CRIPTE (developed at the ONERA) 1, which is a general purpose tool dedicated to solve topological problems. More recently, an open-source tool developed by the York university 2 was published to solve topological problems but it is specific to the PWB approach (it is not a general purpose tool).

A similar approach was proposed more recently by [Tait et al., 2011] where transmission ACS are determined for two configurations: nested reverberation chambers with aperture coupling and weapons bays in a fighter aircraft.

**Selection of approaches for the study of couplings inside cavities from an EMSEC perspective.**

The need to determine coupling in reverberant environments is justified from an EMSEC point of view as explained in the previous chapter. An identified constraint is that the boundaries of the system may not be clearly characterized due to the fact that they may vary in the course of time or because the environment where the inner elements (like PCBs, cables, etc.) are integrated is unknown a priori. Such a configuration can correspond to a computer chassis inside which additional elements are added over time, or when its internal components (cables, etc.) are not mechanically constrained nor constant over the time. One can also consider the early design stage of equipment (when its exact future casing shape is not determined yet) but the coupling between elements need to be assessed anyway. Additionally, the selected method needs to take into account an electromagnetic interference that couples to the cavity through apertures. By determining couplings, we meant that probability density functions (PDFs) of coupled currents and voltages need to be determined, with randomized boundary conditions. From these values, on can determine the probability:

• that a radio front end becomes unavailable (remind section I.1.3.2.3).

• to propagate information by means of polyglot signals (see section I.1.3.1.4).

• that a covert channel is created by taking benefit from the susceptibility to conducted EMI of an electric component or a system (see section I.1.3.3).

• that a voltage/current correlated to sensitive information reaches a threshold at which the confidentiality of information is not guaranteed anymore (TEMPEST threat, see section I.1.3.1).

• that a component/system stops to work properly during an IEMI (section I.1.3.2).

• etc.

To solve these problems, the plane waves superposition representation of the field inside an overmoded cavity (see section I.2.5) will be favourably used. If the boundary conditions are randomized, thus for each draw a new set of eigenmodes are enforced. By considering numerous random boundary conditions, and through an ensemble average of the field inside the cavity, the cavity of interest can be considered as overmoded. Thus, all the methods, that require that regime, apply.

**Table of contents :**

Introduction

**I State of the art **

I.1 ElectroMagnetic SECurity (EMSEC)

I.1.1 Introduction

I.1.2 EMSEC at component level

I.1.3 EMSEC at system level

I.1.4 EMSEC at building level

I.1.5 Conclusion

I.1.6 Need to study reverberant environments

I.2 Couplings to and inside reverberant environment

I.2.1 Introduction

I.2.2 Electromagnetic couplings through apertures

I.2.3 Electromagnetic couplings within a cavity

I.2.4 Electromagnetic field to conductors couplings

I.2.5 Cavity regimes

I.2.6 Conclusion

I.2.7 Selection of approaches for the study of couplings inside cavities from an electromagnetic

security (EMSEC) perspective

**II Design and assessment of computer chassis models **

II.1 Design of computer chassis models

II.1.1 Introduction

II.1.2 The simulation model

II.1.3 Design and production of a computer chassis mock-up

II.1.4 Design and production of devices to measure couplings

II.1.5 Conclusion

II.2 Comparison between the two models for two configurations

II.2.1 Introduction

II.2.2 Electric field amplitude and couplings assessment with the two models .

II.2.3 Applicability of the simulation model to determine statistics

II.2.4 Conclusion

**III Application of the Random coupling model **

III.1 Application of the Random Matrix Theory to determine couplings inside microwave cavities

III.1.1 Introduction

III.1.2 The Random Matrix Theory (RMT)

III.1.3 The random coupling model (RCM)

III.1.4 Set up of the random coupling model (RCM) for the two configurations of interest

III.2 Determination of the impedance/admittance of an aperture

III.2.1 Introduction

III.2.2 First estimation

III.2.3 Second estimation

III.2.4 Impedance of an aperture between two regions

III.2.5 Mapping of the aperture impedance to the RCM

III.3 Numerical computation of the aperture admittance

III.3.1 Introduction

III.3.2 Numerical error estimation

III.3.3 Integration of Grad mn over kx

III.3.4 Integration of Brad,ms mn over kx and ky

III.3.5 Computation of Brad mn by means of the Hilbert transform

III.3.6 Impedance and admittance

III.4 Monte-Carlo simulations

III.4.1 Introduction

III.4.2 Elements statistics of the normalized impedance matrix ξ

III.4.3 RCM implementation

III.4.4 Statistics of the elements of the normalized impedance matrix

III.4.5 Relation between the variance of ii, ij and α

III.4.6 Chaoticity of the random normalized impedance matrix

III.4.7 Statistical quantity of interest for EMSEC

III.4.8 First RCM application

III.4.9 Conclusion

III.5 Measurements

III.5.1 Introduction

III.5.2 Use of a small stirrer to generate a chaotic environment

III.5.3 Methods to determine the parameters of the RCM

III.5.4 First comparison: monopoles

III.5.5 Second comparison: printed circuit boards

III.5.6 Effects of absorbers on the magnitude of induced currents

III.5.7 Coupling between the aperture and the internal ports

III.5.8 Conclusion

**Conclusion **

**IV Appendix **

**A Tables of the mock-up deformations **

**B Admittance of an aperture**

**C Derivation of the Random coupling model **

C.1 Introduction

C.2 Derivation of the RCM in two dimensions

C.3 Statistical representation

C.4 Radiation impedance as a deterministic quantity

C.5 Generalisation to the multiple port case

**D Determination of the antenna factor from experimental data, and of the electric field from the antenna factor **

D.1 Determination of the electric field strength at a distance D from an antenna using its antenna factor AF

D.2 Determination of the antenna factor with two identical antennas

**E Résumé des travaux **

E.1 Introduction

E.2 La sécurité électromagnétique (SECEM)

E.3 Méthodes d’étude des couplages dans les cavités

E.4 Établissement et évaluation de deux modèles de châssis d’ordinateur

E.5 Le modèle de couplages aléatoires

E.6 Conclusion

E.7 Travaux futurs