Contenidos - tsc.unex.estsc.unex.es/~tabo/ROG/ROG_tema1_02.pdf · Radiación y ondas guiadas – J. M. Taboada 10 Escuela Politécnica Universidad de Extremadura Maxwell’s equations

Radiación y ondas guiadas – J. M. Taboada 1

Escuela Politécnica U

niversidad de Extrem

adura Contenidos

� Tema 1. Fundamentos de radiación electromagnética: � Fundamentos de radiación.

� Distribuciones de corriente.

� Teorema de Poynting. � Potenciales retardados.

� Radiación de una fuente elemental.

� Campos radiados por una antena. � Propiedades del campo radiado: campo cercano, intermedio y lejano.




adura Contenidos

� Tema 2. Conceptos básicos de antenas. � Tipos de antenas.

� La antena como elemento circuital: parámetros de impedancia.

� Coeficiente de reflexión y relación de onda estacionaria. � Diagrama de radiación. Directividad. Ganancia y eficiencia.

� Polarización.

� Ancho de banda. � La antena en recepción.

� Fórmula de Friis: propagación en espacio libre.

� Ecuación de alcance radar. � Ruido captado por una antena.




adura Contenidos

� Tema 3. Antenas de hilo. � Integral de radiación.

� Dipolos eléctricos. Monopolo sobre plano de tierra.

� Teoría de imágenes. � Dipolos paralelos a plano conductor.

� Otras antenas de hilo.

� Acoplamientos mutuos entre antenas. � Antenas Yagi.




adura

UNIT 1 Fundamentals of electromagnetic radiation




adura Antennas

� An antenna is a usually metallic device for radiating or receiving radio waves.

� It is the transitional structure between free-space and the transmission line. � The transmission line is used to transport the electromagnetic energy from the

transmitting or to the receiver device.

� Properties of a good antenna:

� Radiation efficiency. � Radiation pattern.

� Transmission line matching.

Tx Transmitter

Waveguide

Antenna

Free-space spherical wavefronts

Rx Receiver

Waveguide

Antenna

Incoming plane wavefronts




adura Current distribution on a thin wire antenna.




adura Electromagnetic radiation mechanism

a) During the first quarter of wavelength the electric current accumulates positive charge in the upper arm and negative in the lower arm. The circuit is closed throughout the displacement currents (field lines of force).

b) During the next quarter the current decreases, generating fields lines on the opposite direction, which pushes forward the previous lines.

c) After the first half of period the net charge on the dipole is null, which forces the field lines to unite together to form closed loops




adura Electromagnetic radiation mechanism

E � �( ) Re j tt e ��e E




adura Maxwell’s equations. Fields:

E: Electric field intensity (V/m) H: Magnetic field intensity (A/m)

D: Electric flux density (Coulombs/m2)

B: Magnetic flux density (Weber/m2) Sources:

J: electric current density

Ji : impressed current

�: electric charge density Constitutive parameters:

�: electric permittivity

�: magnetic permeability

Faraday law: Ampere law:

Gauss law: Magnetic flux contin:

Continuity equation:

Constitutive relations:

dielectric loss factor (usually it is function of �) dielectric permittivity in DC




adura Maxwell’s equations. The current J is composed of an impressed (excitation known

current), a conduction current:

and the total current is the latter :

Impressed current

Conductivity

Displacement current

Conduction current

Displacement reactive current

Displacement dissipative current

Conduction and friction losses in dielectric




adura Maxwell’s equations

� Sometimes it is convenient to introduce a fictitious magnetic current density M.

� Magnetic currents are useful as equivalent sources that replace complicated electric fields in some problems.




adura Boundary conditions

General case: penetrable object Perfect electric conducting (PEC) object

1 2ˆ ( ) 0n �E E

1 2ˆ ( ) 0n � �D D1 2ˆ ( ) 0n �H H

1 2ˆ ( ) 0n � �B B

1 2ˆ ( ) 0n �E E

1 2ˆ ( ) sn �� D D1 2ˆ ( ) sn �H H J

1 2ˆ ( ) 0n � �B B

1 1,tanˆ 0 0n � � �E E

1 1,norˆ s sn � �� D D

1 1,tanˆ ˆs sn n � � � H J H J

1 1,norˆ 0 0n � � � �B B

Conductividad finita. Sin cargas ni corrientes inducidas en interfaz.

Conductividad finita. Cargas y/o corrientes inducidas en interfaz.

Conductividad infinita (PEC).

Normally Js = 0; �s=0




adura Poyinting vector. Conservation of power.

� The complex Poyinting vector represents the complex power density in W/m2 at a point

� Complex power flowing out from a closed surface S surrounding the antenna:

P’source = Pradiated + Pstored magn. + Pstored elect. + Pdissipated

with

because




adura Poyinting vector. Conservation of power.

� We are particularly interested in the real power (the part of the source power that can be radiated)

Psource = Pradiated + Pdissipated




adura Current distribution

� Obtaining the current distribution on antennas or scatterers (in magnitude and phase) is one of the most complex problems in electromagnetics

� It is given by the boundary conditions and the excitation (incident field or voltage source) of the problem. Depends on geometry, material, feed point, etc.

� Nowadays it is addressed using numerical techniques such as the method of moments (MoM). Depending on the electric size of the problem it could imply solving matrix systems with millions or hundreds of millions of unknowns.

� We are going to suppose that the induced currents J are known. The goal here is to develop procedures for finding the radiated fields by an antenna or scatterer based on Maxwell’s equations




adura Potentials

� The radiation problem consists of solving for the fields that are created by a source current distribution (in the further denoted by J instead of Ji). The source currents may represent either actual or equivalent currents. How to obtain these currents will be discussed in later courses (it constitutes a hot research topic in computational electromagnetics). For the moment, suppose we have the source current distribution J and we wish to determine the fields E and H.

� It is possible to obtain directly from the source currents by integration. � However it is much simpler to do it in two steps:

� 1. Find the auxiliary functions (vector potentials, or potenciales retardados) by integration.

� Magnetic vector potential A

� Scalar potential �

� 2. Find the radiated fields by differentiation.




adura Potentials

� Magnetic vector potential A. From the continuity of the magnetic flux law:

� Scalar potential . From the Faraday law:

� So the fields can be obtained in terms of the potential functions. We now discuss the

solution for the potential functions A and .

ad

because

because




adura Potentials

� The Ampere law can be written in terms of the potential vectors

� Lorentz condition (fixing the divergence of A):

� We lead to the magnetic vector potential wave equation (or Helmholtz equation for A)

because because

� and � can be replaced by the equivalent ones in the case of lossy dielectrics

By the uniqueness theorem, if we reach a solution that fulfills Maxwell’s equations, this is the real only solution to the problem.




adura Potentials

� The same procedure can be applied to the Gauss law:

� This is the wave equation or Helmholtz equation for the scalar potential

because (Lorentz condition)




adura Calculation of fields from potentials

� Once we have the potentials we can calculate the fields:

� The electric field can be also obtained using only the vector potential

This is important because in this way we can only work in terms of electric current J, without explicitly considering the electric charge ��

� Relation between fields in absence of fonts. From Ampere law with J=0: �

because (Lorentz condition)




adura Radiation from an elemental source

� The simplest radiation element is an infinitesimal lineal current density element Jz with length dl in an isotropic medium.

� The problem has spherical symmetry, so we shall use the spherical coordinate system.

� Solution

x

y

z

zJ

zI J ds�

dl

ds dv dlds�

because the source is a point

Bessel spherical differential equation




adura Radiation from an elemental source

� Derivation of constants:

� From the vector potential we obtain the fields:

x

y

z

zJ

zI J ds�

dl

ds2

0C � because it represents an incoming wave

Medium intrinsic impedance




adura Radiation from antennas

� A real current distribution is composed of infinite current elements J defined inside of infinitesimal volumes dV placed at points r’.

� The total potential is given by superposition:

Volume Surface Wire antenna

( ')J r

'�r r

r

'r

'dv




adura Antenna radiated field regions

� An inspection of the radiated fields of the infinitesimal dipole

reveals that the space surrounding the

antenna can be subdivided into three regions:

� Near-field region (r < �): predomination of 1/r3 terms

� Intermediate region

� Far-field region (r >> �): predomination of 1/r terms




adura Near-field region (r < �)

� The 1/r3 terms predominates the field:

� The time-average power density reduces to zero because E and H are in quadrature:

� The fields are reactive and quasistatic




adura Intermediate-field (or Fresnel) region (kr > 1)

� The terms that were predominant for the near region become smaller

� As r increases, E and H approach time-phase, which is an indication of formation of time-average power flow in the outward (radial) direction (radiation phenomenon)

� The radiation fields predominate over the reactive fields. The angular field distribution is dependent on the distance from the antenna.

� E has some component in the radial direction (it is named a cross field)

E




adura Far-field region (r >> �)

� Far field region, or Fraunhofer region

� Important notes:

� Fields E and H in the far field region are orthogonal each other and orthogonal to the radial propagating direction, thus locally behaving like a plane wave.

� The magnitude of the E and H fields are related by the medium intrinsic impedance.

� Even for an infinitesimal source we have a directional behavior of , implying that it is impossible to obtain a totally isotropic antenna!

� Complex power density is real indicating dissipated power and it is travelling away from the source and decreasing as 1/r2, typical for spherical progressive waves. This power is named radiated power, and the fields radiated fields.

Outgoing power density:

r

E




adura Radiation of power

� The energy of an harmonically excited antenna dissipates due to: (i) a finite (Ohmic) resistance felt by the charge carriers in the metal wire and (ii) loss of energy due to radiation of e.m. waves.

� This so-called radiation loss occurs due to the fact that the oscillation eventually creates time-dependent electric fields at remote distances, which must then be accompanied by magnetic fields that vary according to Maxwell’s equations.

� At large enough distance these fields transform into plane waves which are free-space solutions of the wave equation. If the dipole oscillation would be suddenly switched off, those far-away fields, or simply far fields, would continue to propagate since they carry energy that is stored in the fields themselves and has been removed from the energy originally stored in the charge distribution we have been starting out with.

� On the contrary, the so-called near-field zone corresponds to the instantaneous electrostatic fields of the dipole, which do not contribute to radiation but return their energy to the source after each oscillation cycle or when the source is turned off (reactive power).




adura Antenna radiated field regions: summary

Far field condition for an antenna: D: maximum antenna dimension




adura Far-field approximation

� We are in the far-field region when

� Approximations

for the amplitude term

for the phase term

( ')J rr

'r





� Approximated vector potential:

� Magnetic field:

� Electric field:

E and H far-field relation:

TEM wave

negligible terms 1/r2

r

'r





� Far-field electric and magnetic radiated fields. Vector expressions:

� Far-field electric and magnetic radiated fields. Scalar expressions:

x y

z

r̂




adura Antenna far-field radiation properties

� The far-field radiated fields of an antenna must fulfill the following aspects:

� The dependency of E and H with r is that of the spherical wave

� E and H depend on � and � because the spherical wave is non homogeneous

� The spherical radiated wave locally behaves like a plane wave

� The E and H fields do not have radial components

� The wave energy density decreases with 1/r2

in a lossless medium:

x y

z

�

� �̂

�̂

r̂




adura Antenna displacement

� Antenna in the coordinate origin:

� Antenna displaced to a point rc

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Contenidos - tsc.unex.estsc.unex.es/~tabo/ROG/ROG_tema1_02.pdf · Radiación y ondas guiadas – J. M. Taboada 10 Escuela Politécnica Universidad de Extremadura Maxwell’s equations