Eddy currents generated in a system of two cylindrical conductors by a transverse alternating magnetic field

In the paper, using the Bubnov–Galerkin method coupled with the separation of variables method, the eddy currents in a system of two cylindrical conductors placed in a transverse homogeneous magnetic field sinusoidally varying with time are investigated. The power losses caused by these currents are also determined. By dint of computer calculations, the graphs of these losses are traced.     

Computing the electromagnetic field of the system of arbitrarily positioned conductors in horizontally stratified multilayer medium

This paper presents a novel computational methodology, based on the finite element technique, for the analysis of electromagnetic field due to system of arbitrarily positioned current‐carrying conductors in horizontally stratified multilayer medium, having arbitrary number of layers with different characteristics (including air). Each soil layer is horizontally unbounded, homogenous and isotropic, whereas conductors can penetrate different layers and extend into the air. The effect of the stratified multilayer medium is taken into account by using the originally developed fixed image method. Complete electromagnetic coupling between grounding system conductors (satisfying thin‐wire approximation) is taken into account, whereas attenuation and phase shift effects are approximated. The electric and magnetic field in stratified multilayer medium are computed from the scalar electric and vector magnetic potentials, using the said fixed image method and approximations to the attenuation and phase shift effects. Copyright © 2014 John Wiley & Sons, Ltd.     

Power losses in the system of two hollow cylindrical conductors placed in transverse alternating magnetic field

Contents The paper presents a method which makes it possible to determine power losses in the system of two long, parallel, cylindrical and hollow conductors placed in transverse magnetic field changing sinusoidally. — The idea is to apply the finite element method in the region with current and the separation of variables in the region without current. The conditions of the vector potential continuity and the tangential component continuity of the magnetic induction vector were taken into account in the set of equations determined by the Bubnov-Galerkin method. — On the basis of the relations obtained, numerical calculations were performed and power losses in the system defined.

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Leistungsverluste in einem aus zwei hohlen Leitern bestehenden und sich in einem querliegenden Wechselfeld befindenden System

Übersicht Im Beitrag wird eine Methode vorgestellt, die die Ermittlung von Leistungsverlusten in einem aus zwei langen parallelen zylindrischen hohlen Leitern bestehenden System, das sich in einem querliegenden Wechselfeld befindet, ermöglicht. — Die Idee der vorgeschlagenen Methode beruht auf einer Anwendung der Methode der finiten Elemente im Strombereich und der Methode der Variablentrennung im stromlosen Bereich. In der Struktur des Gleichungssystems, das von der Bubnov-Galerkin-Methode bestimmt wird, hat man Stetigkeitsbedingungen für das Vektorpotential und für die Tangentenkomponente des Vektors der magnetischen Induktion berücksichtigt. — Auf Grundlage der hergeleiteten Abhängigkeit werden numerische Berechnungen durchgeführt und die Leistungsverluste im betrachteten System ermittelt.

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Symbols A vector potential (complex r.m.s. value) - A z-component ofA (complex r.m.s. value) - B the magnetic induction of input (complex r.m.s. value) - d distance between two successive points of the boundary - E electric field intensity vector (complex r.m.s. value) - e ₁ metric coefficient - H magnetic field intensity vector (complex r.m.s. value) - I current (r.m.s. value) - Imaginary unit - Imaginary unit - l boundary of the region with current - l ^h boundary of the region with current approximated by broken line - P complex Poynting vector - P Joule's power losses - r _c radius of the conductor - r _w radius of the hollow - S boundary of the region - T _ijk finite element area - w half a distance between centres of the conductors - x,y,z rectangular coordinates - , ,z bipolar coordinates - z ^* conjugate complex number ofz - _c value of the coordinate on the boundary of the conductor - region - ^h region approximating the region - ^e finite element region - _i, _j, _k function of finite-element shape - permeability - conductivity - pulsation - basis function - 3.141593.... - derivative in the normal external direction with respect to the region boundary - ² scalar Laplacian Indices I region without current - II region with current - b relative values - e single element - i, j, k suffices of vertices of triangular finite-element - L number of discretization points of the region - M number of discretization points on the boundary - N number of the sum terms in the series expansion - V number of finite elements     

Conservation of energy of the transient electromagnetic field in solid conductors

Contents The calculation of transient electromagnetic fields in solid conductors becomes more important in the future because of the increasing demands on system design under EMC-constraints. The switching of currents in a system of conductors is especially crucial because a transient wave is stimulated in the non-conducting environment. The energy balance for the switching of an exciting current shows that the integration of the dissipated power over the infinite time interval is equal to the energy stored in the magnetic field in the conducting space before switching plus the energy that flows through the conductor's surface during the transient process. The energy difference between the initial field before and immediately after the switching is radiated into the free space by a transient wave that is excited at the moment of switching. This relation is proved for the switching of a continuous current in an elementary coaxial line and the shielding of the transient field of a magnetic dipole of arbitrary orientation by a conducting and permeable hollow sphere.

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Energiebilanz des transienten elektromagnetischen Feldes massiver Leiter

Übersicht Die Berechnung transienter elektromagnetischer Felder in Leitersystemen gewinnt unter dem Gesichtspunkt der elektromagnetisch verträglichen Gestaltung von Systemen zunehmend an Bedeutung. Kritisch sind insbesondere Schaltvorgänge, bei denen in der nichtleitenden Umgebung eine transiente elektromagnetische Welle angeregt wird. Eine Energiebilanz zeigt, daß die Verlustenergie, die nach dem Schalten eines Stromes in den Leitern umgesetzt wird, gleich der Energie ist, die vor dem Schalten im stationären magnetischen Feld gespeichert war, vermehrt um die Energie, die während des transienten Vorgangs aus der nichtleitenden Umgebung über die Leiteroberfläche in die Leiter einströmt. Die Differenz zur gesamten Energie des ursprünglichen stationären Feldes wird durch die transiente Welle abgestrahlt. Diese Relationen werden für das Schalten eines Stromes im Koaxialkabel und für einen kugelförmigen Schirm im Feld eines magnetischen Dipols beliebiger Orientierung nachgewiesen.

     

One-dimensional cylindrical diffusion of electromagnetic fields,part I

Contents We discuss magnetic fieldsB _z (r, t) andB (r, t) diffusing into homogeneous conducting circular cylinders of radiusr ₀ with boundary conditionsB _z (r ₀,t) orB (r ₀,t) proportional tot ⁿ. Laplace-transforms are used. The main difficulty is their inversion for larger values ofn. The procedures can be strongly simplified by the introduction of certain polynomials. They have very remarkable properties. They are also helpful for many applications. If one wants to calculate the dissipated Joule-heat for instance, one needs certain infinite sums related to the eigenvalues of the problems. These infinite sums can easily be evaluated with the help of the polynomials mentioned. The corresponding plane problems are also considered in order to show that these polynomials are the cylindrical analogues of Bernoulli- and Euler-polynomials. The relations between our polynomials and Fourier-Bessel-expansions are the same as those between Bernoulli- and Euler-polynomials and Fourier-expansions. Finally hollow cylinders are discussed, too. The results are similar but more complicated than for full cylinders.

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Eindimensionale zylindrische Diffusion elektromagnetischer Felder, Teil I

Übersicht Wir behandeln magnetissche FelderB _z (r, t) undB (r, t), die in homogene leitfähige Kreiszylinder mit dem Radiusr ₀ diffundieren mit der Randbedingung, daß fürr=r ₀ die Felder proportionalt ⁿ sind. Dabei wird die Laplace-Transformation benutzt, wobei deren Inversion für größere Werten sehr umständlich ist. Durch die Einführung bestimmter Polynome kann das Vorgehen sehr erleichtert werden. Diese Polynome haben bemerkenswerte Eigenschaften und können für viele Anwendungen sehr nützlich sein. Will man z.B. die dissipierten Energien berechnen, so benötigt man dazu gewisse unendliche Summen der Eigenwerte der Probleme, die mit Hilfe der genannten Polynome leicht berechnet werden können. Die Behandlung der analogen ebenen Probleme zeigt, daß diese Polynome das zylindrische Analogon der Bernoulli-und Euler-Polynome sind. Die Beziehungen zwischen unseren Polynomen und Fourier-Bessel-Reihen sind dieselben wie die zwischen Bernoulli-und Euler-Polynomen und Fourier-Reihen. Abschließend werden auch Hohlzylinder behandelt. Die Ergebnisse sind ähnlich, jedoch erheblich komplizierter als für Vollzylinder.

     

One-dimensional cylindrical diffusion of electromagnetic fields, Part II

Contents This paper continues a previous one [1]. It discusses a magnetic fieldB _z (r,t) diffusing into a homogeneous conducting cylinder (of radiusr ₀). The difference between the two papers is that different boundary conditions are applied. The boundary condition now is an integrated one, the magnetic flux within a coaxial hollow cylinder (of radiusR ₀>r ₀) being proportional tot ⁿ (actually a more general problem is discussed). As in the previous paper the solution can be simplified by the introduction of certain polynomials, which are very useful and which have interesting properties. They are generalisations of the polynomials defined in [1]. The corresponding plane problem is discussed again, too. The polynomials defined in this case are related to generalisations of Bernoulli- and Euler-polynomials.

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Eindimensionale zylindrische Diffusion elektromagnetischer Felder, Teil II

Übersicht Die Arbeit stellt die Weiterführung einer vorhergehenden Arbeit [1] dar. In ihr wird die Diffusion eines Magnetfeldes in einen homogenen leitfähigen Zylinder (Radiusr ₀) diskutiert. Der Unterschied zwischen beiden Arbeiten liegt in den Randbedingungen. Hier ist die Randbedingung eine integrale. Der magnetische Fluß innerhalb eines koaxialen Hohlzylinders (RadiusR ₀>r ₀) ist proportional zut ⁿ (tatsächlich wird ein allgemeineres Problem behandelt). Wie in [1] kann die Lösung durch die Einführung bestimmter Polynome vereinfacht werden, die bemerkenswerte Eigenschaften haben und sehr nützlich sind. Sie stellen Verallgemeinerungen der in [1] eingeführten Polynome dar. Wie früher wird auch hier das analoge ebene Problem behandelt. Die dabei auftretenden Polynome hängen mit Verallgemeinerungen von Bernoulli-und Euler-Polynomen zusammen.

     

Semi-analytical approach for electromagnetic field diffusion using hertz potential

(Paper written in French) This paper contains two parts: In the first part the Hertz vectorial potential is introduced and its possible use in solving electromagnetic diffusion problems common in electrical engineering is considered. Links between this potential and other commonly used potentials are established, and a formulation procedure using three vectorial equations is proposed. This approach has several advantages due to its simplicity and the fact that it requires only a few values for computation. This method is next applied efficiently to the interaction in the static regime between a line conductor crossed by a sine-shaped current and a finite-thickness metallic plate. In the second part, a specific field quadrature method is employed in integral form. This allows the confirmation of the methods accuracy and good numerical stability.     

Skin effect in a bifilar lead consisting of cylindrical conductors

The electromagnetic field of a bifilar lead consisting of two thick cylindrical conductors carrying an alternating current is investigated by using the Bubnov-Galerkin method. The electrodynamic forces acting on these conductors and their inner impedances are examined. By digital computations the graphs of the forces and impedances are given for conductors with equal radii.     

Influence of the two-dimensional skin- and proximity-effect on electromagnetic field in the conductors systems carrying nonsinusoidal current

Contents The paper present the influence of the skin- and proximity-effect on Joule power losses in the conducto systems placed in a homogeneous medium and carry opposite periodic nonsinusoidal currents. Theoretical calculation and numerical computation hav been carried out by means of the Bubnov-Galerkin method using numerical calculations following the finite elements method, for the conducting region and separation of variables for the air region. On the basis of numerical computation, graphs of the relative power losses for anarbitrary current with a waveform periodically variable in time, an for different skin- and proximity-effect parameters are plotted.

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Einfluß des zweidimensionalen Skin- und Proximity-Effektes auf das elektromagnetische Feld eines Leiter-systems mit nichtsinusoidalen Strömen (In Englisch)

übersicht Der Beitrag stellt den Einfluß der Skin- und Proximity-Effekte auf die Jouleschen Verluste eines Leiter-systems dar. Das System, das sich in einem homogenen Raum befindet, wird von entgegengesetzten, nichtsinusoidalen, periodischen Strömen durchflossen. — Die theoretischen Betrachtungen und die numerischen Berechnungen wurden in Anlehnung an die Bubnov-Galerkin-Methode (in ihrer numerischen Version der Methode der finiten Elemente für Bereiche des Leiters) in Verbindung mit der Methode der Variablentrennung für den Luftraum durchgeführt. — Auf der Grundlage der numerischen Berechnung wurden Diagramme der relativen Verluste für einen willkürlich ausgewählten Strom mit zeitlich periodischem Verlauf und für verschiedene Parameter des Skin- und Proximity-Effektes gezeichnet.

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Symbols A vector potential (complex r.m.s. value) - A z-component ofA (complex r.m.s. value) - B magnetic induction vector (complex r.m.s. value) - d distance between two successive points of the boundary - E electric field intensity vector (complex r.m.s. value) - e single element - H magnetic field intensity vector (complex r.m.s. value) - i, j, k vertices numbers of the considered finite element - I current (r.m.s. value) - I _o direct component of current - imaginary unit - k harmonic numbers - KR skin-effect parameter - L number of discretization points of the region - M number of discretization points on the boundary - N number of the sum terms in the series expression - N _p finite element shape function - P complex Poynting vector - P Joule power losses - R radius of the conductors - S boundary of region - T period - WR proximity-effect parameter - ^c value of the coordinate on the boundary of the conductor - ^l basis function - magnetic permeability - angular frequency - ¹ basis angular frequency - conductivity - conducting region - ^e finite element region - 3.141593...... - ² scalar Laplacian - finite element area - x, y, z rectangular coordinates - , ,z bipolar coordinates - respect to the region boundary     

瞬态电磁场辐射敏感度测试系统

随着科学技术的发展,各种电子设备大量使用,电磁干扰成为关注的焦点,要进行电磁干扰实验首先必须建立模拟电磁场.本文介绍了可以模拟核电磁脉冲的瞬态电磁场辐射敏感度测试系统,论述了该测试系统建立的必要性、功能、系统组成及使用方法.     

A numerical study of the efficiency of a cylindrical electromagnetic shield

Electromagnetic shields are widely used for protecting electrical equipment and cable lines from the influence of natural and artificial electric and magnetic fields. A 3D mathematical model of a solid cylindrical shield for protection from the effect of an alternating electromagnetic field produced inside a cable has been proposed. The model has been constructed using the finite-element method in the ANSYS HFSS software. Two electromagnetic-field sources have been considered: noise in the inner conductor (cable core) and a coaxial circuit, in which the shield is an outer working conductor. The electromagnetic-field distribution over the cable is described by the Maxwell equations, which are transformed into the Helmholtz vector equation with respect to the electric-field component so that the finite-element method may be used. The electromagnetic parameters of the considered media are constant and isotropic. The shielding properties have been studied for a copper cylindrical shield in the frequency range of 1–100 MHz. As a result of the mathematical solution, the distributions of electromagnetic-field intensities in the model have been obtained. To determine the shielding properties of a cable, the transfer impedance and shielding attenuation have been used. The adequacy of the model and technique for determining the efficiency of the screening has been verified by means of comparison between the obtained data and results of the analytical model.     

Non-symmetrical shaped conductors located in a time-harmonic TM field

Contents In the work the method providing to determinate the power losses of long conductor of an arbitrary polygonal cross-section, placed in a time-harmonic transverse magnetic (TM) field. This method rests on the connection of the Bubnov-Galerkin method in its numerical version performed by means of finite element method for internal region with the method of division of variables for external region. — On the basis of theoretical considerations the numerical calculations were performed for several chosen cases of shaped conductors, on the basis of which the plots of Joule power losses in those conductors were made.

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Die in einem senkrechten harmonischen Magnetfeld (TM-Feld) lokalisierten nicht symmetrischen Profilleiter

Übersicht Im Beitrag wurde eine Methode zur Ermittlung der Leistungsverluste in einem langen Leiter mit einem beliebigen vielblätterigen Querschnitt angegeben. Der Leiter befindet sich in einem harmonischen senkrechten Magnetfeld (TM-Feld). Die Methode verbindet die Bubnov-Galerkin-Methode (es geht hier um nummerische Realisierung dieser Methode unter Verwendung der Methode finiter Elemente für den inneren Bereich) und die Variablentrennung-Methode (für den äußeren Bereich). — Auf der Grundlage theoretischer Erwägungen hat man numerische Berechnungen für einige gewählte Leiterprofile durchgeführt und die Joule-Diagramme der Leistungsverluste erhalten.

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List of main symbols A z-component of the vector potentialA (complex r.m.s. value) - B magnetic induction vector (complex r.m.s. value) - B _t =A/n tangential component of the magnetic induction vector (complexs r.m.s. value) - E electric field vector (complex r.m.s. value) - i, j, k numbers of vertices of the considered finite element - H magnetic field intensity vector (complex r.m.s. value) - Imaginary unit - l boundary of the region II - l ^h boundary of the region II approximated by broken line - P complex Poynting vector - P Joule's power losses - r ₀ radius - S boundary of the region - x, y, z rectangular coordinates - r, ,z cylindrical coordinates - =3.141593 ... - z ^* conjugate complex number ofz - basis function - magnetic permeability - pulsation - N _i ,N _j ,N _k function of finite element shape - conductivity - region - ^h region approximating the region - ^e finite element region - ² scalar Laplacian - finite element area     

The influence of multi-layer ground on the electromagnetic field of an overhead power transmission line in the presence of buried conductors

Contents The present work investigates the two dimensional, quasi stationary, electromagnetic field of a faulted power transmission line in the presence of a buried pipeline, of mitigation wires and of a multi-layer ground. The related diffusion equation has been numerically solved by using the Finite Element Method (FEM). Using FEM results and Faraday's law, magnetic vector potential, as well as the voltages induced across the buried pipeline and remote earth, are calculated. Parametric analysis has shown that there is a significant influence of the depth and resistivity of the first ground layer, of the resistivities of the different ground layers and of the configuration of mitigation wires on the electromagnetic field and on the voltages induced across the buried pipeline and remote earth.

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Der Einfluß der mehrschichtigen Erde auf das elektromagnetische Feld einer Übertragungsleitung in der Anwesenheit von unterirdischen Leitern

Übersicht In der vorliegenden Arbeit wird das zweidimensionale, quasi stationäre, elektromagnetische Feld einer mit einem Fehler behafteten Übertragungsleitung in der Anwesenheit einer unterirdischen Rohrleitung, mehreren Schtzleitungen und einer mehrschichtigen Erde untersucht. Die entsprechende Diffusionsgleichung wird numerisch mit Hilfe der Finiten Elementen Methode (FEM) berechnet. Anschließend werden mit Hilfe der Ergebnisse der FEM und der Faraday schen Gleichung das magnetische Vektorpotential und die zwischen der Rohrleitung und der fernen Erde induzierten Spannungen berechnet. Untersuchungen zeigten, daß für das Feld und die zwischen Rohrleitung und der fernen Erde induzierten Spannungen folgende Parameter von großer Bedeutung sind: Die Dichte der ersten Erdschicht, die spezifischen Widerstände der Erdschichten und die Anordung der Schutzleitungen.

     

Numerical investigations of electromagnetic processes in a solid cylindrical shield

A mathematical model is suggested to describe the processes in a solid cylindrical shield in protection against an alternating magnetic field. The model is constructed with respect to the complex amplitude of magnetic vector potential. Since magnetic field lines are in a plane perpendicular to the axis of a cylindrical shield, the problem becomes two-dimensional. The electromagnetic parameters of the considered media are constant and isotropic. The plates at which the magnetic potential is set are the source of the magnetic field. A distribution of real and imaginary components of the complex amplitude of magnetic potential is described by four differential equations in the conducting medium and by two equations in the dielectric one. An equality of magnetic potential at both sides of the interface is predetermined at the interfaces. The Robin boundary condition provides equality of the magnetic vector potential to zero at an infinite distance from the shield. The obtained differential equation system supplemented with the boundary conditions can be numerically solved by the finite elements method using the Galerkin method. As a result, distributions of magnetic potential and magnetic field intensity in the absence and presence of a shield are determined; shielding attenuation is then calculated. It is found that, with increasing shield thickness and noise frequency, the efficiency of electromagnetic shielding is increased. The adequacy of the suggested model and technique of determination of the shielding efficiency is corroborated by comparison with the results of an analytical model for a copper cylindrical shield.     

On the accuracy of a 3-D infinite element for open boundary electromagnetic field analysis

Contents This paper describes an application of the three dimensional infinite element for modelling open boundary conditions in the finite element analysis of electromagnetic field problems. The infinite element has four nodes and is compatible with conventional cuboids. It is shown that the infinite element serves not only to obtain higher calculation precision in the interior—finite element region, but also gives very good results in the exterior region.

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Qualitätsanalyse von 3-D-Infiniten-Elementen für unberandete elektromagnetische Felder

Übersicht In der Arbeit wurde die Verwendung der 3-D-Infiniten-Elemente zur Modellierung der elektromagnetischen Feldprobleme in unberandeten Gebieten dargestellt. Das infinite Element hat vier Knoten und ist völlig kompatibel mit konventionellen Rechteck-Prismenelementen. Es wurde gezeigt, daß die infiniten Elemente die Genauigkeit der Lösung verbessern und sehr gute Ergebnisse sowohl im Innen-wie Außengebiet liefern.

     

Assessment of torque components in brushless permanent-magnet machines through numerical analysis of the electromagnetic field

For the calculation of torque in brushless (BL) alternating current motors a local method is proposed, based on the Maxwell stress theory and the filtered contributions due to the harmonics of the magnetic vector potential in the motor air gap. By considering the space fundamental field only, the method can efficiently estimate the average synchronous torque for a variety or motor topologies, including concentrated winding designs. For BL direct current motor analysis a global method is introduced, based on the virtual work principle expressed in terms of energy components in various motor regions. The method leads to simplifications in the average torque calculation and enables the direct identification of the cogging and ripple components. The mathematical procedures have been validated against experiments and other numerical techniques.     

无网格法及其在电磁场计算中的应用展望

先从与FEM相比较的角度，以操作流程的方式介绍了无网格法(MLM)的原理和特点，指出了MLM和FEM之间是一般和特殊的关系。然后介绍了MLM的发展历史、研究队伍和应用现状；经过对发表在电磁计算领域的MLM文献的全面综述，可以看到MLM在电磁计算中的研究才刚起步。最后介绍了MLM的局限性和尚待解决的关键技术问题，并对MLM在电磁计算中的潜在研究课题和应用前景进行了展望。     

Eddy-currents in conductors system with a.c.

Contents A numerical method of calculation of the eddycurrent induced within the system of parallel conducting bars of infinite length by a.c. flowing through these is presented. The integral equation approach is applied, which makes it possible to obtain an aproximate solution of the problem considered. As an example the eddy-current density within rectangular bars is calculated using digital computer.

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Wirbelströme in einem Leitersystem bei sinusförmigen Wechselströmen

Übersicht In der Arbeit wird eine Methode zur Berechnung der Wirbelströme in parallelen und unendlichen langen unmagnetischen Leitern bei sinusförmiger Zeitabhängigkeit angegeben, u. zw. wird eine Integralgleichungsmethode angewandt. Als Beispiel werden die Wirbelströme in Leitern mit rechteckigen Querschnitten berechnet.

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List of Symbols A vector potential - I current - J current density - conductivity - _ij Kroneeker delta - ₀ 4·10^–7 H/m-permeability of vacuum - angular frequency     

On the electromagnetic scattering of a chaff cloud

Scattering characteristics of a chaff are investigated, using a cloud of wire dipoles. The influence of mutual coupling between the dipoles is taken into account. The cloud is studied in terms of the resonant conditions. Algorithms to specify the stochastic distribution of the dipole positions and orientations are constructed. Finally, the possibility of tracing an obscure target behind the cloud is examined.