electromagnetic wave propagation ec 442

ELECTROMAGNETIC WAVE PROPAGATION

EC 442 Prof. Darwish Abdel Aziz

CHAPTER 6

LINEAR WIRE ANTENNAS INFINITESIMAL DIPOLE

INTRODUCTION

Wire antennas, linear or curved, are some of the

oldest, simplest, cheapest, and in many cases the

most versatile for many applications.

May, 2015 3 Prof. Darwish

INFINITESIMAL DIPOLE 1 - INTRODUCTION

An infinitesimal linear wire is positioned symmetrically at the origin of the coordinate system as shown in Figure (6-1).

Figure 6-1 Infinitesimal dipole


INFINITESIMAL DIPOLE

• Although infinitesimal dipoles are not very practical, they are

used to represent capacitor – plate ( also referred to as top-

hat-loaded) antennas.

• In addition, they are utilized as building blocks of more

complex geometries.

• The end plates are used to provide capacitive loading in

order to maintain the current on the dipole nearly uniform. • Since the end plates are assumed to be small, their radiation

is usually negligible. May, 2015 5 Prof. Darwish

INFINITESIMAL DIPOLE 2 – CURRENT DISTRIBUTION

• The wire, in addition to being very small , is very

thin .

• The spatial variation of the current is assumed to be

constant and it current element is given by

Where, .

• The remaining two equations are unchanged from their static (non time-varying) form:


INFINITESIMAL DIPOLE 3 – RADIATION EQUATIONS

• Since

• So,

, and Where, and and


INFINITESIMAL DIPOLE 3 – RADIATION EQUATIONS

• So , and,


INFINITESIMAL DIPOLE 4 – AUXILIARY VECTOR POTENTIAL FUNCTION

• So the electric vector potential components are:

• While the magnetic vector potential components are:

, and


INFINITESIMAL DIPOLE 5 – THE RADIATED FIELD COMPONENTS

• The Magnetic Field Components can be found as follows:



and • The Electric Field Components can be found as follows:



So ,



and and


INFINITESIMAL DIPOLE 6 – THE RADIAL AND TRANSVERSE POWER DENSITY

• For the infinitesimal dipole, the complex Poynting vector can be written using (6-6a) - (6-6b) and (6-8a) - (6-8c) as

Whose radial and transverse components are given,

respectively, by May, 2015 14 Prof. Darwish

INFINITESIMAL DIPOLE 7 – THE RADIAL POWER

• The complex power moving in the radial direction is obtained by integrating (6-9) – (6-10b) over a closed sphere of radius . Thus it can be written as

which reduces to May, 2015 15 Prof. Darwish

INFINITESIMAL DIPOLE 8 – THE REACTIVE POWER

• The transverse component of the power density does not

contribute to the integral. Thus (6-12) does not represent

the total complex power radiated by the antenna.

• Since , as given by (6-11b), is purely imaginary, it will not

contribute to any real radiated power.

• However, it does contribute to the imaginary (reactive)

power which along with the second term of (6-12) can be

used to determine the total reactive power of the antenna.


INFINITESIMAL DIPOLE 9 – THE TOTAL OUTWARDLY RADIAL POWER

• The reactive power density, which is most dominant for

small values of , has both radial and transverse

components.

• Equation (6-11b), which gives the real and imaginary

power that is moving outwardly, can also be written as



Where

From (6-12)



• It is clear from (6-15) that the radial electric energy must

be larger than the radial magnetic energy.

• For large values of , the reactive

power diminishes and vanishes when .


INFINITESIMAL DIPOLE 10 – RADIAN DISTANCE AND RADIAN SPHERE

for infinitesimal dipole, as represented

by (6- 6a) - (6- 6c) and (6- 8a) - (6- 8b), are valid everywhere

(except on the source itself). An inspection of these

equations reveals the following:

• At a distance ,

which is referred to as the radian distance,

the magnitude of the first and second terms within the

brackets of (6-6c) and (6-8a) is the same.



Also at the radian distance the magnitude of all three terms

within the bracket of (6 – 8b) is identical; the only term that

contributes to the total field is the second, because the first

and third terms cancel each other.

• At distances less than the radian distance ,

the magnitude of the second term within the brackets of

(6 - 6c) and (6 – 8a) is greater than the first term and begins

to dominate as .

o



For (6-8b) and , the magnitude of the third term

within the brackets is greater than the magnitude of the

first and second terms while the magnitude of the second

term is greater than that of the first one; each of these terms

begins to dominate as .

The near-field region, is defined as the region

, and the energy in that

region is basically imaginary (stored).



• At distances greater than the radian distance ,

The first term within the brackets of (6-6c) and (6-8a) is

greater than the magnitude of the second term and begins to

dominate as .

For (6-8b) and , the first term within the brackets is

greater than the magnitude of the second and third terms

while the magnitude of the second term is greater than that of

the third; each of these terms begins to dominate as .



The intermediate - field region is defined as the region

The far- field region is defined as the region

, and the energy in that region

is basically real (radiated).

• The radian sphere is defined as the sphere with radius

equal to the radian distance .



• The radian sphere defines the region within which the

reactive power density is greater than the radiated power

density.

For an antenna, the radian sphere represents the volume

occupied mainly by the stored energy of the antenna’s

electric and magnetic fields.

Outside the radian sphere the radiated power density is

greater than the reactive power density and begins to

dominate as .



• The radian sphere can be used as a reference, and it defines

the transition between stored energy pulsating primarily in

the direction [represented by (6-10b)] and energy

radiating in the radial direction [represented by (6-10a); the

second term represents stored energy pulsating inwardly

and outwardly in the radial direction].

• Similar behavior, where the power density near the antenna

is primarily reactive and far away is primarily real, is

exhibited by all antennas, although not exactly at .


INFINITESIMAL DIPOLE 11 – NEAR FIELD REGION

• An inspection of (6-6a)- (6-6b) and (6-8a)- (6-8c) reveals that for or they can be reduced in much simpler form and can be approximated by


INFINITESIMAL DIPOLE 11 – NEAR FIELD REGION

• The components, are in time- phase but

they are time- phase quadrature with the

component ; therefore there is no time-average power

flow associated with them. This is demonstrated by forming

the time- average power density as

which by using (6-16a)- (6-16d) reduces to


INFINITESIMAL DIPOLE 12 – INTERMEDIATE FIELD REGION

• As the values of begin to increase and become greater

than unity, the terms that were dominant for become

smaller and eventually vanish. For moderate values of

the components lose their in-phase condition and

approach time-phase quadrature. Since their magnitude is

not the same, in general, they form a rotating vector whose

extremity traces an ellipse.



• At these intermediate values of , components

approach time-phase, which is an indication of the

formation time-average power flow in the outward (radial)

direction (radiation phenomenon).

• As the values of become moderate , the field

expression can be approximated again but in a different

form. In contrast to the region where , the first term

within the brackets in (6-6b) and (6-8a) becomes more

dominant and the second term can be neglected.



• The same is true for (6-8b) where the second and third

terms become less dominant than the first.

• Thus we can write for


INFINITESIMAL DIPOLE 13 – FAR - FIELD REGION

• Since (6-19a) - (6-19d) are valid only for values of ,

then will be smaller than because is inversely

proportional to where is inversely proportional to .

• In a region where , (6-19a) - (6-19d) can be simplified

and approximated by



• The ratio of to is equal to

where

The components are perpendicular to each

other, transverse to the radial direction of propagation, and

the variations are separable from of variations.



• The shape of the pattern is not a function of the radial

distance , and the fields form a Transverse ElectroMagnetic

(TEM) wave whose wave impedance is equal to the intrinsic

impedance of the medium.

• As it will become even more evident, this relationship is

applicable in the far-field region of all antennas of finite

dimensions.


INFINITESIMAL DIPOLE 14 – FAR FIELD RADIATED COMPONENTS

• The far field components of (6-20a) - (6-20c) can also

be derived using the procedure outlined and

relationships developed in chapter-5 of auxiliary

vector potential functions.

• The far field radiated components using the radiation

equations can be

written as:


INFINITESIMAL DIPOLE 14 – FAR FIELD RADIATED COMPONENTS


INFINITESIMAL DIPOLE 15 – POWERE DENSITY AND RADIATION RESISTANCE

• The input impedance of an antenna, which consists of real and imaginary parts as discussed in Chapter- 4 (Fundamental Parameters of Antenna).

• For a lossless antenna, the real of the input impedance was designated as radiation resistance, through which the radiated power is transferred from the guided wave to the free space wave.

• To find the input resistance for a lossless antenna, it is required to find the time average poynting vector as



• The total radiated power in the radial direction is

obtained by integrating (6-23e) over a closed sphere of

radius . Thus it can be written as:



• Since the antenna radiates its real power through the

radiation resistance, for the infinitesimal dipole it can be written that

• For free space medium, , where is the intrinsic impedance, so


INFINITESIMAL DIPOLE 16 – DIRECTIVITY

• As was shown before, the average power density of the infinitesimal dipole is given by (6-23e) as

• As was discussed in Chapter- 4 (Fundamental Parameters

of Antenna), the radiation intensity can be obtained from



• The maximum value of the radiation intensity occurs at and it is equal to

• The real power radiated by the infinitesimal dipole is

given by (6-24e) as



• As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), the directivity is given as

• As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), for lossless antenna, the relation between the directivity and the maximum effective aperture area is given as


electromagnetic wave propagation ec 442

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