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Basic Antenna Theory

- Application

- A. Vector Analysis
- 1. Scalar Product

- 2. Vector Product

- 3. Circulation

Circulation of

Closed path

- 4. Flux

- 5. Gradient

(Rect. Coord.)

Del Operator

- 6. Divergence

ROC of a Field in Direction of Field

- 7. Curl

Rect. Coord.

Curl is a measure of fields ability to produce

torque

8. Laplacian

Rect. Coord.

9. Helmholtz Theorem

A vector is completely defined if its DIV Curl

are specified if the field vanished as r?h (or

if normal component is specified over some closed

surface) Note, we specified that DIV Curl are

ROCs are related to sources of the field.

Thus a vector field is specified by its vector

scalar source.

10. Divergence Theorem

10. Stokes Theorem

B. Maxwells Equations 1. Gauss Law

Line of Charge

For a Closed Surface

Apply Divergence Theorem

Maxwell Eqn. (Gauss Law for Points)

2. Amperes Law

For a very long straight wire,

lines encircle conductor

In this case,

3. Potential

Conservative Field

And

Grad. of a scalar

From

To find V from

a collection of charges

?

For general situations

where

Now, you can get from

4. Equation of Continuity

Current

Closed Surface Around One Plate

Current Entering Closed Surface

For I out of closed surface

By Divergence Theorem

Equation of Continuity

Returning to Amperes Law

In General Maxwell Equation

Current density is composed of two parts

5. Gauss Law for Magnetics

6. Faradays Law

Apply Stokes Thm

7. M.E. and Sources of Fields

8. Magnetic Vector Potential

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9. Poynting Vector and Power

Outward Power Flow

From Stored Magnetic Electric Energies

Instantaneous Poynting Vector

C. Radiation

(Assume sinusoidal time variation)

1. Total

the gradient of a scalar

Scalar Potential from Charges

2. Wave Equations

From

Get

And

(sometimes called ß)

Looks messy, but note we still have not

specified yet. If we choose

Lorentz Gauge Condition

(which is really just the equation of continuity

in disguise!)

we get

And for single source notation

Solutions to Wave Eqns. Are Found by

Considering a Point Source (called a free-space

Greens Function

observation pt.

source pt.

is the wave eqn. for G

(this you can show with lots of effort)

We then deduce

convolution integral

Potential integrals are solutions to wave

equation!!

3. Elementary Source

We will now solve a most simple antenna and use

it as a building block (in our computer modeling)

to solve complicated antennas.

Short Electric Dipole, Constant (uniform) I

How short is short ? Why is it called a

Dipole? How can you achieve uniform I ?

dipole moment

Find from using sph.

coordinates.

1/r 1/r2

1/r3 Far Out Close In Real Close In

3 Regions

Far Field

In-phase terms Ratio of ? 1/r2 power variation

Magnitude of plane wave

Plane wave

Spherical wave!

Radially outward and variation No loss of

power, just spreads out

, N-S , E-W

Strongest at Equator Zero at Poles

Up close looks like a plane wave

Near and Fresnel

Usually lumped together into near field

90o out of phase with Energy oscillates

in out and is capacitive

(Near fields tied to J, P. Far fields tied to

displacement current)

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Near Antenna

E H out of phase H in phase with I E in phase

with Q on ends

Far from Antenna

Never returns E in phase with H

P

Einduction Eradiation are 180o out of

phase Hinduction Hradiation are 90o out of

phase Why?

- D. Antenna Concepts

1.

Antennas are field warpers which transform

transmission line fields into spherical waves

with prefered directions and vice

versa. Antennas made of wires (linear antennas)

have currents and charges on them and if you know

or can guess I Q, you can find the far out

stuff via potential integrals (solutions to the

wave equation).

2.

If you dont know I or Q, you have lots of

hard work to do, because all you have is a

source function and conducting surface where E

is perpendicular to surface and Etangential is

zero. You then have to write a boundary value

equation to force Etan 0 at the conductors by

writing Etan in terms of where K is a

Greens function. The problem then is to find I

which is unfortunately inside the .

3.

Once you have E, H you can use S to get the

total power radiated over a sphere and define

4.

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Patterns are usually plotted as a cross section

of the 3D surface in a plane. Often we choose

the planes as perpendicular cuts through the main

beam (maximum lobe).

E Plane Contains axis of main beam and

vector

H Plane Contains axis of main beam and

vector

Toroidal Shaped

Could plot fields too,

Gain and Directivity are dependent on . A

maximum (constant) value is often specified

6.

In general

for measurements

Antenna

(same r)

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Since

Far field

GT, GR are often off-frequency

What about near field coupling? Tough! Got to

use network concepts.

Note how super-simple idealized the short

dipole is how complex the solution was. We

could do the same set-up for the next simplest

antenna, a ?/2 dipole but would find we would

have to integrate the assumed current to find

. Anything more complex than that we would

have to know the current and be able to

integrate it.

There is another possibility We could treat

complex structures as a collection of short

dipoles of different but uniform currents!

Looks simpler and it is but you must know or

guess I( l). IF you cant do this, you must

develop an integral equation for I.

- D. Integral Equation Formulation

The typical linear antenna is a cylindrical

dipole

L/2

2a

Z0

Gap is fed by

-L/2

field or Vo v. generator

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Apply surface boundary condition

With Greens Function Notation,

with

Reduce volume to surface by assuming perfect

conductor applying axial symmetry it is OK to

consider just one value of

Pocklingtons Eqn.

Look at Kernel

It has a singularity of third order at and

the integration is over a surface, thus the

integral diverges, and Pocklingtons Eqn. cannot

be used for numerical analysis. This does not

make the integral equation meaningless. If the

THIN WIRE APPROXIMATION is applied

And observation points are taken along a line on

the cylinder surface so that

We get

Pocklingtons Thin Wire Equation

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