Lecture 8.Ppt

Antenna EngineeringAntenna EngineeringEC 544EC 544

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

LECTURE 8LECTURE 8Rectangular Rectangular

ApertureAperture

Rectangular ApertureRectangular Aperture

In practice, the rectangular aperture is probably In practice, the rectangular aperture is probably the most common microwave antenna. Because the most common microwave antenna. Because of its configuration, the rectangular coordinate of its configuration, the rectangular coordinate system is the most convenient system to express system is the most convenient system to express the fields at the aperture and to perform the the fields at the aperture and to perform the integration.integration.The three most common and convenient The three most common and convenient coordinate positions used for the solution of an coordinate positions used for the solution of an aperture antenna are :aperture antenna are :(1) The aperture lies on the y-z plane.(1) The aperture lies on the y-z plane.(2) The aperture lies on the x-z plane.(2) The aperture lies on the x-z plane.(3) The aperture lies on the x-y plane.(3) The aperture lies on the x-y plane.

Rectangular ApertureRectangular Aperture

Rectangular ApertureRectangular ApertureWith Ground PlaneWith Ground Plane

Uniform Distribution on an Infinite Ground Plane

1 – The Aperture Field1 – The Aperture FieldThe field over the aperture is assumed to be The field over the aperture is assumed to be constant and given by :constant and given by : -a/2 ≤ x’≤ a/2-a/2 ≤ x’≤ a/2 -b/2 ≤ y’≤ b/2-b/2 ≤ y’≤ b/2

Where EWhere Eoo is a constant. is a constant.

The task is to find the fields radiated by it, HPBW, The task is to find the fields radiated by it, HPBW, the side lobe levels, and the directivity. the side lobe levels, and the directivity. To accomplish these, the equivalent will be formed To accomplish these, the equivalent will be formed first. first.



2 – Equivalent Problem2 – Equivalent Problem- To form the equivalent, a closed surface is chosen which - To form the equivalent, a closed surface is chosen which extends from – extends from – фф to to фф on the x – y plane. on the x – y plane.-By applying the equivalence and the image theory, the By applying the equivalence and the image theory, the equivalent problem becomes…equivalent problem becomes…


NNrr = 0 , L = 0 , Lrr = 0 = 0


Principle Planes Principle Planes

Uniform Distribution on an Infinite Uniform Distribution on an Infinite Ground PlaneGround Plane


• DirectivityDirectivity

The directivity for the aperture can be found fromThe directivity for the aperture can be found from

The analytical details using this procedure, especially The analytical details using this procedure, especially the integration to compute the radiated power (pthe integration to compute the radiated power (pradrad), ), are more cumbersome.are more cumbersome.

Because the aperture is mounted on an infiniteBecause the aperture is mounted on an infinite ground plane, an alternate and much simpler method ground plane, an alternate and much simpler method can be used to compute the radiated power. can be used to compute the radiated power.


DirectivityDirectivity The average power density is first formed using the The average power density is first formed using the fields at the aperture, and it is then integrated over fields at the aperture, and it is then integrated over the physical bounds of the opening.the physical bounds of the opening.

Assuming that the magnetic field at the aperture is Assuming that the magnetic field at the aperture is given bygiven by

The radiated power reduces toThe radiated power reduces to


DirectivityDirectivityThe radiation intensity can be determined using the The radiation intensity can be determined using the far field radiation asfar field radiation as

The maximum radiation intensity U|The maximum radiation intensity U|maxmax occurs towards occurs towards θθ equal zero, and it is equal to equal zero, and it is equal to


DirectivityDirectivityThus the directivity is equal toThus the directivity is equal to

WhereWhere


Rectangular ApertureRectangular ApertureUniform Distribution In free SpaceUniform Distribution In free Space

Uniform Distribution in Free Space

In this case, the aperture is not mounted on an infinite In this case, the aperture is not mounted on an infinite ground plane. The field distribution is given byground plane. The field distribution is given by

2 – Equivalent Problem2 – Equivalent Problem

To form the equivalent, a closed surface is chosen To form the equivalent, a closed surface is chosen which again extends from – which again extends from – фф to + to + фф on the x-y plane. on the x-y plane. Over the entire surface JOver the entire surface Js s and Mand Ms s are formed. The difficulty are formed. The difficulty encountered in this problem is that both Jencountered in this problem is that both Js s and Mand Ms s are not zero are not zero out side the opening, and expression for them are not known out side the opening, and expression for them are not known there. The replacement of the semi-infinite medium to the left there. The replacement of the semi-infinite medium to the left of the boundary (- z) by an imaginary electric or magnetic of the boundary (- z) by an imaginary electric or magnetic conductor only eliminates one or the other current densities conductor only eliminates one or the other current densities (J(Js s and Mand Mss) but not both. Because the fields outside the ) but not both. Because the fields outside the opening are not known a priori. An approximate equivalent will opening are not known a priori. An approximate equivalent will be adopted. The usual and most accurate relaxation is to be adopted. The usual and most accurate relaxation is to assume that both Eassume that both Eaa and H and Haa (and in turn M (and in turn Mss and J and Jss) exist over ) exist over the opening but are zero outside it.the opening but are zero outside it.



NNrr = 0 , L = 0 , Lrr = 0 = 0


3- Radiation Equations3- Radiation Equations

Uniform Distribution in Free Space3- Radiation Equations3- Radiation Equations

Uniform Distribution in Free Space4 – Far Field Components


Principle Planes Principle Planes

To find the beamwidths and the angle at which the To find the beamwidths and the angle at which the maximum of the side lobe occurs, it is usually maximum of the side lobe occurs, it is usually assumed that theassumed that the (1 + cos (1 + cos θθ) term is a much slower) term is a much slower

varying function than terms. varying function than terms.

This an approximation, and it is more valid for large This an approximation, and it is more valid for large aperture and for angles near the main maximum. So aperture and for angles near the main maximum. So the expressions that were derived for uniform the expressions that were derived for uniform distribution on an infinite ground plane can be used.distribution on an infinite ground plane can be used.


Beamwidths and Side Lobe Levels


To derive an exact expression for the directivity of To derive an exact expression for the directivity of this aperture would be a very difficult task. Since the this aperture would be a very difficult task. Since the patterns of the apertures (on an infinite ground patterns of the apertures (on an infinite ground plane and on free space) are nearly the same, plane and on free space) are nearly the same, especially at the main lobe, their directivities are especially at the main lobe, their directivities are almost the same.almost the same.

DirectivityDirectivity

Rectangular ApertureRectangular ApertureTMTM1010

Mode Mode Distribution Distribution Ground PlaneGround Plane

TMTM1010 Mode Distribution Ground PlaneMode Distribution Ground Plane

In practice, a commonly used aperture In practice, a commonly used aperture antenna is that of rectangular waveguide antenna is that of rectangular waveguide mounted on an infinite ground plane. At the mounted on an infinite ground plane. At the opening, the field is usually approximated opening, the field is usually approximated by the dominant TEby the dominant TE10 10 – mode.– mode.

2 – Equivalent Problem2 – Equivalent Problem • To form the equivalent, a closed surface is chosen To form the equivalent, a closed surface is chosen which which extends from – extends from – фф to to фф on the x – y plane. on the x – y plane.

• By applying the equivalence and the image theory, the By applying the equivalence and the image theory, the equivalent problem becomesequivalent problem becomes



3- Radiation Equations3- Radiation Equations


4- 4- Far Field Far Field ComponentsComponents


• The expressions for the beamwidths and side lobe level in the E – plane are identical to those of uniform aperture field. However those for the H – plane are more complex, and a simple procedure is not available.

• Computation for the HPBW, FNBW, FSLBW, FSLMM in the E – and H – planes are shown graphically in the following figures.

Beamwidths and Side Lobe LevelsBeamwidths and Side Lobe Levels


E-plane: Uniform / TEE-plane: Uniform / TE1010 mode Ground Plane mode Ground Plane


H-plane: Uniform / TEH-plane: Uniform / TE1010 mode Ground Plane mode Ground Plane


Directivity and Aperture Directivity and Aperture EfficiencyEfficiencyThe directivity of this aperture is found in the The directivity of this aperture is found in the same manner as that of the uniform distribution same manner as that of the uniform distribution aperture. Using the aperture electric field and aperture. Using the aperture electric field and assuming that the magnetic field is related to the assuming that the magnetic field is related to the electric field by the intrinsic impedance electric field by the intrinsic impedance ηη, the , the radiated power can be found as…radiated power can be found as…

The radiation intensity can be determined using the The radiation intensity can be determined using the far field radiation asfar field radiation as


The maximum radiation intensity occurs at The maximum radiation intensity occurs at θθ = 0 = 0oo, and , and it is given byit is given by

Thus the directivity is equal toThus the directivity is equal to


Aperture EfficiencyAperture Efficiency

Aperture Efficiency Aperture Efficiency

Aperture Effective AreaAperture Effective Area

Beam Efficiency (BE)Beam Efficiency (BE)

Circular ApertureCircular Aperture

Lecture 8.Ppt

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