lecture 7.ppt

Antenna TheoryAntenna TheoryEC 544 EC 544

Prof.Dr. Darwish Abdel AzizProf.Dr. Darwish Abdel Aziz

Chapter Chapter 77

Radiation Integrals and Radiation Integrals and Auxiliary Potential Auxiliary Potential

FunctionsFunctions

Radiation Integrals and Radiation Integrals and Auxiliary Potential Auxiliary Potential

FunctionsFunctions

• In the analysis of radiation problems, the usual In the analysis of radiation problems, the usual procedure is to specify the sources and then require procedure is to specify the sources and then require the fields radiated by the sources. This is in contrast the fields radiated by the sources. This is in contrast to the synthesis problem where the radiated fields to the synthesis problem where the radiated fields are specified, and the sources are required to be are specified, and the sources are required to be determined.determined.

• It id a very common practice in the analysis It id a very common practice in the analysis procedure to introduce auxiliary functions, known as procedure to introduce auxiliary functions, known as vector potentials, which will aid in the solution of the vector potentials, which will aid in the solution of the problems.problems.

• The most common vector potential functions are the The most common vector potential functions are the magnetic vector potential A and electric vector magnetic vector potential A and electric vector potential F.potential F.

Auxiliary potential Auxiliary potential FunctionsFunctions


• Although the electric and magnetic field intensities Although the electric and magnetic field intensities (E and H) represent physically measurable (E and H) represent physically measurable quantities, among most engineers the potentials are quantities, among most engineers the potentials are strictly mathematical tools.strictly mathematical tools.

• The introduction of the potentials often simplifies The introduction of the potentials often simplifies the solution even though it may require the solution even though it may require determination of additional functions.determination of additional functions.



The Magnetic vector The Magnetic vector potentialpotential


• The magnetic vector potential A is useful in solving The magnetic vector potential A is useful in solving for the EM field generated by a given harmonic for the EM field generated by a given harmonic electric current J. The magnetic flux B is always electric current J. The magnetic flux B is always solenoid, that is, solenoid, that is,

• The magnetic vector potential A is useful in solving The magnetic vector potential A is useful in solving for the EM field generated by a given harmonic for the EM field generated by a given harmonic electric current J. The magnetic flux B is always electric current J. The magnetic flux B is always solenoid, that is, solenoid, that is,

• Therefore, it can be represented as the curl of Therefore, it can be represented as the curl of another vector because it obeys the vector identity another vector because it obeys the vector identity • Therefore, it can be represented as the curl of Therefore, it can be represented as the curl of another vector because it obeys the vector identity another vector because it obeys the vector identity

Thus we defineThus we define



Substituting into Maxwell’s curl equationSubstituting into Maxwell’s curl equation

Which will be reduced to…Which will be reduced to…



Which can also be written asWhich can also be written as

So…So…

SinceSince

So….So….



From Maxwell’s equations, we can writeFrom Maxwell’s equations, we can write

Which can be reduced toWhich can be reduced to

The curl of A was defined before, and the divergence The curl of A was defined before, and the divergence of A can be defined so as to simplify the previous of A can be defined so as to simplify the previous equation, asequation, as

Finally the inhomogeneous Helmholtz equation can be Finally the inhomogeneous Helmholtz equation can be obtainedobtained

Once A is known, HOnce A is known, HAA and E and EAA can be found. can be found.



The Electric vector The Electric vector potentialpotential


Thus we defineThus we define

Substituting into Maxwell’s curl equationSubstituting into Maxwell’s curl equation

Which will be reduced toWhich will be reduced to



Solution of the Inhomogeneous Solution of the Inhomogeneous Vector Potential Wave EquationVector Potential Wave EquationSolution of the Inhomogeneous Solution of the Inhomogeneous Vector Potential Wave EquationVector Potential Wave Equation

The solution to the vector wave equation for the The solution to the vector wave equation for the magnetic vector potential A magnetic vector potential A

can be written ascan be written as


ForFor

Far–Field RadiationFar–Field RadiationFar–Field RadiationFar–Field Radiation

Far–Field RadiationFar–Field RadiationFar–Field RadiationFar–Field Radiation

Radiation EquationsRadiation EquationsRadiation EquationsRadiation Equations

SummarySummary

Transformation into Spherical Transformation into Spherical CoordinatesCoordinates

Rectangular Aperture / surfaceRectangular Aperture / surface

Field Equivalence Principle Field Equivalence Principle Huygens’ PrincipleHuygens’ Principle

• The field Equivalence : “is a principle by which actual sources, such as an antenna

and transmitter, are replaced by equivalent sources.”

• The field Equivalence : “is a principle by which actual sources, such as an antenna

and transmitter, are replaced by equivalent sources.”

It is a more rigorous formulation of Huygens’ principle which states that “each point on a primary wave front can be considered to be a

new source of a secondary spherical wave and that a secondary wave front can be constructed as the envelope of these secondary spherical waves.” and it is based on the uniqueness theorem

It is a more rigorous formulation of Huygens’ principle which states that “each point on a primary wave front can be considered to be a

new source of a secondary spherical wave and that a secondary wave front can be constructed as the envelope of these secondary spherical waves.” and it is based on the uniqueness theorem

The Uniqueness Theorem: states that “a field in a lossy region is uniquely specified by the

sources within the region plus the tangential components of the electric field over the boundary, or the tangential

components of the magnetic field over the boundary, or the former over part of the boundary and the latter over the rest

of the boundary.”

The Uniqueness Theorem: states that “a field in a lossy region is uniquely specified by the

sources within the region plus the tangential components of the electric field over the boundary, or the tangential

components of the magnetic field over the boundary, or the former over part of the boundary and the latter over the rest

of the boundary.”


By the By the equivalence principleequivalence principle, the fields outside an imaginary , the fields outside an imaginary closed surface are obtained by placing over the closed surface closed surface are obtained by placing over the closed surface suitable electric- and magnetic- current densities which satisfy suitable electric- and magnetic- current densities which satisfy the boundary conditions. The current densities are selected so the boundary conditions. The current densities are selected so

that the fields inside the closed surface are zero and outside that the fields inside the closed surface are zero and outside they are equal to radiation produced by the actual sources. they are equal to radiation produced by the actual sources.

Thus the technique can be used to obtain the fields radiated Thus the technique can be used to obtain the fields radiated outside a closed surface by sources enclosed within it.outside a closed surface by sources enclosed within it.

By the By the equivalence principleequivalence principle, the fields outside an imaginary , the fields outside an imaginary closed surface are obtained by placing over the closed surface closed surface are obtained by placing over the closed surface suitable electric- and magnetic- current densities which satisfy suitable electric- and magnetic- current densities which satisfy the boundary conditions. The current densities are selected so the boundary conditions. The current densities are selected so

that the fields inside the closed surface are zero and outside that the fields inside the closed surface are zero and outside they are equal to radiation produced by the actual sources. they are equal to radiation produced by the actual sources.

Thus the technique can be used to obtain the fields radiated Thus the technique can be used to obtain the fields radiated outside a closed surface by sources enclosed within it.outside a closed surface by sources enclosed within it.

The equivalence principle is developed by considering an actual radiating source, which electrically is represented by current densities J1 and M1

A closed surface S is chosen, which encloses the current densities (J1 and M1 ).

The primary task will be to replace the original problem by an equivalent one which yields the same fields E1 H1 outside S (within V2).



An equivalent problem is formed by removing the original sources An equivalent problem is formed by removing the original sources JJ11

and and MM11, and, and assuming the existence of fields E and H inside S, and assuming the existence of fields E and H inside S, and

the fields the fields EE11 and and HH11 outside of S.outside of S.

For these fields to exist within and outside S, they must satisfy the For these fields to exist within and outside S, they must satisfy the boundary conditions on the tangential electric and magnetic field boundary conditions on the tangential electric and magnetic field components.components.Thus on the imaginary surface S there must exist the equivalence Thus on the imaginary surface S there must exist the equivalence sourcessources


The equivalent sources radiate into an unbounded space. They are The equivalent sources radiate into an unbounded space. They are said to be equivalent only within Vsaid to be equivalent only within V22, because they produce the original , because they produce the original

fields (Efields (E11, H, H11) ) only outside S.only outside S.

Since the fields E, H within S can be anything (this is not the region Since the fields E, H within S can be anything (this is not the region interest), it can be assumed that they are zero. interest), it can be assumed that they are zero. In that case the equivalent problem reduces to what is known as In that case the equivalent problem reduces to what is known as Love’s equivalent, whereLove’s equivalent, where

Love’s Equivalence Love’s Equivalence principle produces a null field within the principle produces a null field within the imaginary surface S.imaginary surface S.

Since the value of the E = H = 0 within S can not be disturbed Since the value of the E = H = 0 within S can not be disturbed if the properties of the medium within it are changed, the if the properties of the medium within it are changed, the medium is assumed to be replaced by a perfect electric medium is assumed to be replaced by a perfect electric conductor (conductor (σσ = = фф).).

The introduction of the perfect conductor will have an effect on The introduction of the perfect conductor will have an effect on the equivalent source Jthe equivalent source Jss, and it will prohibit the use of the , and it will prohibit the use of the auxiliary vector potentials A, and F, since the current densities auxiliary vector potentials A, and F, since the current densities no longer radiate into an unbounded medium. As the electric no longer radiate into an unbounded medium. As the electric conductor takes its place, the electric current density Jconductor takes its place, the electric current density Jss, which , which is tangent to the surface S, is short circuited by the electric is tangent to the surface S, is short circuited by the electric conductor. Thus the equivalent problem reduces toconductor. Thus the equivalent problem reduces to

Love’s Equivalence Love’s Equivalence principle produces a null field within the principle produces a null field within the imaginary surface S.imaginary surface S.

Since the value of the E = H = 0 within S can not be disturbed Since the value of the E = H = 0 within S can not be disturbed if the properties of the medium within it are changed, the if the properties of the medium within it are changed, the medium is assumed to be replaced by a perfect electric medium is assumed to be replaced by a perfect electric conductor (conductor (σσ = = фф).).

The introduction of the perfect conductor will have an effect on The introduction of the perfect conductor will have an effect on the equivalent source Jthe equivalent source Jss, and it will prohibit the use of the , and it will prohibit the use of the auxiliary vector potentials A, and F, since the current densities auxiliary vector potentials A, and F, since the current densities no longer radiate into an unbounded medium. As the electric no longer radiate into an unbounded medium. As the electric conductor takes its place, the electric current density Jconductor takes its place, the electric current density Jss, which , which is tangent to the surface S, is short circuited by the electric is tangent to the surface S, is short circuited by the electric conductor. Thus the equivalent problem reduces toconductor. Thus the equivalent problem reduces to



Instead of placing a perfect electric conductor (Instead of placing a perfect electric conductor (σσ = = фф) ) within S, a perfect magnetic conductor is introduce. The within S, a perfect magnetic conductor is introduce. The introduction of the perfect magnetic conductor will have introduction of the perfect magnetic conductor will have an effect on the equivalent source Man effect on the equivalent source Mss, and it will prohibit , and it will prohibit the use of the auxiliary vector potentials A, and F, since the use of the auxiliary vector potentials A, and F, since the current densities no longer radiate into an unbounded the current densities no longer radiate into an unbounded medium. As the magnetic conductor takes its place, the medium. As the magnetic conductor takes its place, the magnetic current density Mmagnetic current density Mss, which is tangent to the , which is tangent to the surface S, is short circuited by the magnetic conductor. surface S, is short circuited by the magnetic conductor. Thus the equivalent problem reduces to…..Thus the equivalent problem reduces to…..

Instead of placing a perfect electric conductor (Instead of placing a perfect electric conductor (σσ = = фф) ) within S, a perfect magnetic conductor is introduce. The within S, a perfect magnetic conductor is introduce. The introduction of the perfect magnetic conductor will have introduction of the perfect magnetic conductor will have an effect on the equivalent source Man effect on the equivalent source Mss, and it will prohibit , and it will prohibit the use of the auxiliary vector potentials A, and F, since the use of the auxiliary vector potentials A, and F, since the current densities no longer radiate into an unbounded the current densities no longer radiate into an unbounded medium. As the magnetic conductor takes its place, the medium. As the magnetic conductor takes its place, the magnetic current density Mmagnetic current density Mss, which is tangent to the , which is tangent to the surface S, is short circuited by the magnetic conductor. surface S, is short circuited by the magnetic conductor. Thus the equivalent problem reduces to…..Thus the equivalent problem reduces to…..

In order to use the auxiliary vector potential, where an In order to use the auxiliary vector potential, where an unbounded region is required, the surface of the electric unbounded region is required, the surface of the electric conductor or the magnetic conductor is assumed to be flat conductor or the magnetic conductor is assumed to be flat and extends to infinity. and extends to infinity.

For this geometry, the For this geometry, the Image TheoremImage Theorem can be appliedcan be applied

In order to use the auxiliary vector potential, where an In order to use the auxiliary vector potential, where an unbounded region is required, the surface of the electric unbounded region is required, the surface of the electric conductor or the magnetic conductor is assumed to be flat conductor or the magnetic conductor is assumed to be flat and extends to infinity. and extends to infinity.

For this geometry, the For this geometry, the Image TheoremImage Theorem can be appliedcan be applied


An Aperture mounted on an An Aperture mounted on an Infinite Ground PlaneInfinite Ground Plane

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