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Raghuvir Tomar Antenna Engineering Notes

for Chapter 1 (RadiationFundamentals)Revision 002

Authors Date Place

Prepared by: Raghuvir Tomar 30th

December2015

LNMIITJaipur,India

Reviewed by:

Reviewed by:


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Antenna Engineering Notes for Chapter 1 by R.Tomar Rev 002 ______________________________________________________________________

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i

Revision History

Revision Date Description

001 21s July, 2014 Initial draft

002 30t

December, 2015 Minor typographical mistakes corrected


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Table of Contents

1

FREQUENCY BANDS .................................................................................... 1

1.1 Nomenclature 1 (TRADITIONAL) ........................................................................ 1

1.2

Nomenclature 2 (RADAR-BASED) ...................................................................... 1

2

DEFINITIONS ................................................................................................. 2

2.1 Radiation .............................................................................................................. 2

2.2 Irradiation ............................................................................................................. 2

2.3 Antenna ................................................................................................................ 2

3

WHY/HOW DOES RADIATION TAKE PLACE AT ALL? .............................. 2

4.

CONSTITUTIVE RELATIONSHIPS ............................................................. 3

5.

MAXWELLS’ EQUATIONS ......................................................................... 4

a.

MAXWELL’S FIRST EQUATION............................................................................... 4

b. MAXWELL’S SECOND EQUATION .......................................................................... 5

c. MAXWELL’S THIRD EQUATION ............................................................................. 6

d. MAXWELL’S FOURTH EQUATION .......................................................................... 7

e. Equation of continuity of current ............................................................................... 7

f.

Integral forms of Maxwell’s equations ........................................................................ 7

g. Types of media .................................................................................................... 8

h. How to define ‘wave’ ........................................................................................... 9

6. Time-varying potentials .................................................................................... 10 a. Equations for A and V for the time-varying case ........................................ 11 b. Potential functions for the sinusoidally time-varying case ......................... 13

7. Infinitesimally small dipole (alternating-current element) ......................................... 13 a. Far-field expressions ..................................................................................... 15 b. Radiated Power .............................................................................................. 16 c. Radiation Resistance ..................................................................................... 17

8.

Short antennas (elementary dipole and elementary monopole) .................................. 17 a. Radiation field of the elementary dipole ....................................................... 18 b. Radiation resistance of the elementary dipole ............................................. 19 d. Radiation resistance of the elementary monpole......................................... 19


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9.

Dipoles and monopoles of arbitrary length ............................................................... 19 a. Analysis of center-fed dipole of arbitrary length.......................................... 20

b. Half-wave dipole ............................................................................................ 20 c. Quarter-wave monopole ................................................................................... 22

10.

Unsolved problems ............................................................................................... 22


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1 FREQUENCY BANDS

Two most commonly used frequency-band nomenclatures are given below.

1.1 Nomenclature 1 (TRADITIONAL)

BAND FREQUENCY

RANGE

WAVELENGTH RANGE

Extremely Low Frequency (ELF) 100Km

Very Low Frequency (VLF) 3KHz-30KHz 10Km-100Km

Low Frequency (LF) 30KHz-300KHz 1Km-10Km

Medium Frequency (MF) 300KHz-3MHz 100m-1Km

High Frequency (HF) 3MHz-30MHz 10m-100m

Very High Frequency (VHF) 30MHz-300MHz 1m-10m

Ultra High Frequency (UHF) 300MHz-3GHz 10cm-1m

Super High Frequency (SHF) 3GHz-30GHz 1cm-10cm

Extremely High Frequency (EHF) 30GHz-300GHz 1mm-1cm

1.2 Nomenclature 2 (RADAR-BASED)

BAND LETTER DESIGNATION FREQUENCY

RANGE

WAVELENGTH RANGE

L Band 1GHz-2GHz

S Band 2GHz-4GHzC Band 4GHz-8GHz

X Band 8GHz-12GHz

K u Band 12GHz-18GHz

K Band 18GHz-26.5GHz

K a band 26.5GHz-40GHz

Q Band 33GHz-50GHz

U Band 40GHz-60GHz

V Band 50GHz-75GHz

E Band 60GHz-90GHz

W Band 75GHz-110GHz

F Band 90GHz-140GHzD Band 110GHz-170GHz


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2 DEFINITIONS

Some commonly used pertinent definitions are summarized below.

2.1 Radiation

a) Radiation is a process in which electromagnetic (EM) waves travel throughvacuum or through some other medium containing material.

b) Radiation can be thought of as the phenomenon of transmitting electromagneticenergy via an unbounded medium (e.g., free space). In other words, no guidingstructures (coaxial lines, waveguides, MIC lines, fibre, etc.) have to be present

for the radiation to occur.

c) Radiation is principally caused by two sources: a) time-varying chargedistribution, b) time-varying current distribution. Charge distribution is generally

measured in Coulomb/m3 and is denoted by Current distribution is generally

measured in Ampere/m3 and is denoted by J.

d) Time-varying charge distribution and time-varying current distribution are notindependent quantities (at least for the time-varying case) and are related by the

well-known equation of continuity

.J=-( / t)

This equation, essentially speaking, mathematizes the principle of conservation of charge

which states the following:

The time-rate of decrease of electric charge within a given volume is equal to the net outwardcurrent flow through the closed surface of the volume.

2.2 Irradiation

Irradiation is the process of getting exposed to radiation.

2.3 Antenna

An antenna is the device which converts non-radiating (that is, guided) electromagneticenergy into radiated electromagnetic energy.

3 WHY/HOW DOES RADIATION TAKE PLACE AT ALL?

Radiation can be visualized as the propagation (or moving away) of an effect in all

directions, from the source.

The universe has decided (for some unfathomable reasons) that the disturbance of electric

and magnetic fields due to a moving (or accelerating) electric charge will propagate

away from the charge at the speed of light (c=300,000,000 Km/sec in vacuum). Once thecharge accelerates, the fields need to re-align themselves although they will prefer not to


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have to do so. The re-alignment effect will be felt with greater and greater time-delay aswe go further and further away from the charge, the speed at which the disturbance

travels being equal to c.

4. CONSTITUTIVE RELATIONSHIPS

The two constitutive relationships that govern the electromagnetic behavior of agiven non-conducting material (whether free-space, dielectric medium, or magnetic

medium) are

B=H (1.1)

D=*E (1.2)

In these equations, H is the magnetic field strength in A/m, E is electric field

strength in V/m, B is the magnetic flux density in weber/m2, and D is the electric flux

density in coulomb/m2. is the permeability of the medium and is defined by

=0r (1.3)

where r is the relative permeability of the medium and 0, the permeability of free-space, is given by

0=4π*10-7

Henry/m (1.4))

is the permeability of the medium and is defined by

=0*r (1.5)

where r is the relative permittivity (dielectric constant) of the medium and 0, the

permittivity of free-space, is given by

0=(1/(36*π))*10-9

Farad/m (1.6)

An additional relationship that holds good for imperfect conductors (electricalconductivity not infinite) is

J=E (1.7)

where is the conductivity of the medium, and E is the incident field that causes a surface

current density, J (in A/m2), on the surface of the conductor.

Eq. (1.7) can easily be derived, at least in the case of a rectangular bar made of material with

finite electrical conductivity, by using Ohm’s law.


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5. MAXWELLS’ EQUATIONS

The four Maxwell’s equations are reproduced below. Please note that bold letters

represent vectors and a dot on top represents ( / t). All possible variations of the fourMaxwellian equations are shown. The last few variations shown (in case of each of the

equations) are for the case when e jt type of time-dependence is assumed (a very practically used case).

a. MAXWELL’S FIRST EQUATION

The first Maxwell’s equation is derivable from Faraday’s law

v(t)= - ( t)(1.8)

In eq. (2.1.1),v(t) is the induced emf and is the magnetic flux whose rate of

change with time is producing v(t). We can rewrite equation (1.8) as

(1.9)

where the left hand side (LHS) equates the induced emf to the line integral of the

corresponding electric field (E) and the right hand side (RHS) equates the magnetic fluxto the surface integral of the corresponding magnetic flux density (B). S represents a

two-dimensional surface enclosed by a closed contour C.

Now let us use Stoke’s theorem

S C

dS curlAdl A ).(. (1.10)

where A represents any arbitrary vector. Eq. (1.10) then yields

XE= - ( B/ t) (M1)

which is the first Maxwellian equation in its differential form. We can re-write this

equation in several other forms, as shown below. As already mentioned, the dot on top ofa mathematical quantity represents the time-derivative of that quantity.

.XE= - B (M1.1)

= -( H/ t) (M1.2)

S C

dS Bt

dl E ..


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= -

r ( H/ t) (M1.3)

.= -H(M1.4)

.

= -

rH (M1.5)

= -

r( H/ t) (M1.6)

= - j

rH (M1.7)

= - jB (M1.8)

Note that the last two representations (M1.7 and M1.8) are applicable only

whene jt

type of time-dependence is assumed for both E and H vectors.

b. MAXWELL’S SECOND EQUATION

The second Maxwell’s equation is derivable, after a suitable modification, fromthe Ampere’s law which states that the line integral of magnetic field around any closed

path C is equal to the total current I enclosed by that path. Mathematically speaking, we

write Ampere’s law as

C

I dl H . (1.11)

Let us now use Stoke’s theorem (eq. 1.10) to convert eq. (1.11) into

XH= J (1.12)

where J represents the surface current density (in A/m2) associated with I.

Eq. (1.12) is good enough when interactions between time-varying electric and

magnetic fields can be assumed to be negligible. As frequency increases, this, however,

becomes less and less of a safe bet. Maxwell’s genius lied in suggesting (based onintuitive reasoning which was later verified experimentally) that an additional term be

added to the RHS of eq. (1.12) to account for the fact that time-varying E gives rise to

effects in total H. Eq. (1.12) was thus modified to

XH= ( D/ t)+J (M2)

where the first term on the RHS represents the so-called ‘displacement-current density’

(in A/m2) that was added-in by Maxwell (for an interesting up-to-date view of

displacement current please visit http://en.wikipedia.org/wiki/Displacement_current).

http://en.wikipedia.org/wiki/Displacement_currenthttp://en.wikipedia.org/wiki/Displacement_currenthttp://en.wikipedia.org/wiki/Displacement_current


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Eq. (M2) is the second Maxwellian equation in its differential form. We can write

this equation in several other forms too, as shown below.

.

XH= D + J (M2.1)= ( E/ t) + J (M2.2)

=

r( E/ t) + J (M2.3)

.= E + J (M2.4)

.

=

rE + J (M2.5)

=

r( E/ t) + J (M2.6)

= j

rE + J (M2.7)

= jD + J (M2.8)

Note that the last two representations (M2.7 and M2.8) apply only when e jt

type oftime-dependence is assumed for both E and H vectors.

c. MAXWELL’S THIRD EQUATION

The third Maxwell’s equation is derivable from Gauss’s law which states that thenet electric flux passing through a closed surface S is equal to the total electric charge Q

enclosed by that surface. In other words,

S

QdS D.(1.13)

We use divergence theorem

V S dV DivAdS A )(. (1.14)

to convert (1.13) into


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V V

dV QdV DivD )( (1.15)

In eq. (1.15), represents the volume charge density in C/m3. This equation easily yields

.D= (M3)

which is the third Maxwellian equation in its differential form.

d. MAXWELL’S FOURTH EQUATION

The fourth Maxwell’s equation is derivable by taking divergence on both sides of

(M1). We get

.B= (M4)

by assuming, without any loss of generality, that the constant involved can be assumed to be zero.

Eq. (M4) is the fourth Maxwellian equation in its differential form.

e. Equation of continuity of current

The equation of continuity (derivable by taking divergence on both sides of M2 and

then using M3) is written as follows:

.J=-( / t) (1.16)

which, for e jt

type of time variation, becomes

.J = - j(1.17)

The equation of continuity, essentially speaking, is a manifestation of the

principle of conservation of charge which states the following:

The time-rate of decrease of electri c charge withi n a given volume is equal to the net

outward cur rent f low through the closed sur face of the volume.

f. Integral forms of Maxwell’s equations

By using Stoke’s and Divergence theorems, the four Maxwell’s equations can also be integrated and converted into their equivalent integral representations given below.


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S C

dS Bt

dl E ..(M.I.1)

I dS Dt

dl H

S C

.. (M.I.2)

V S

dV dS D .(M.I.3)

0. S

dS B(M.I.4)

In the above equations, vector dl represents an infinitesimally small length on a

closed path C that is bounding a surface S, vector dS represents an infinitesimally smallarea on the surface S, and dV represents an infinitesimally small volume within the total

volume V bounded by the surface S.

In words, eqs. (M.I.1)-(M.I.4) can be summarized as follows:

a) The electromotive force (emf) around a closed path is equal to the time derivativeof the magnetic flux through any surface bounded by that path.

b) The magnetomotive force (mmf) around a closed path is equal to the total currentI flowing through any surface bounded by the path. This current, in general, is

made up of two components, a fictitious displacement current and a conductioncurrent.

c) The total electric flux through the surface enclosing a volume is equal to the totalcharge contained within the volume.

d) The net magnetic flux emerging through any closed surface is zero.

g. Types of media

The various types of transmission media encountered in practice are briefly discussed below.

1. Homogeneous and non-homogeneous media


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Homogeneous media (e.g., free-space) are those whose properties (permeability, permittivity, and conductivity) do not change with (x,y,z). Inhomogeneous media (e.g.,

human body) are those whose properties do change with (x,y,z).

2. Isotropic and anisotropic media

Isotropic media (e.g., free-space) are those whose properties do not change withdirection. Anisotropic media (e.g., human body) are those whose properties do changewith direction.

For anisotropic materials, the constitutive relationships become (instead of eqs. 1.1and 1.2):

E D (1.18)

H B (1.19)

where [D], [E], [B], and [H] are 3x1 matrices and [] and [] are 3x3 matrices.

3. Linear and non-linear media

Linear media (e.g., free-space) are those whose response to the simultaneous

presence of more than one source can be computed using a linear superposition of themedia responses to individual sources. Non-linear media (e.g., many semiconductors) are

those whose response to more than one source can not be treated as a linearsuperposition of the responses to individual sources.

h. How to define ‘wave’

If a physical phenomenon that occurs at one place at a given time is reproduced at

other places at later times, the time delay being proportional to the space separation fromthe first location, then the group of phenomena is said to constitute a wave.


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6. Time-varying potentials

For a static volume charge distribution (r’), the electric scalar potential V ( r) is given

by

(1.20)

where the integration is carried out over the entire volume over which the charge is

distributed, and R ,r, and r’ are defined in Figure 1.

Figure 1 Definitions

Note that the vector r’ denotes the position vector of the elemental volume over which

the charge is being considered, and is called the ‘source-coordinate’. Vector r, on the

other hand, denotes the position vector of the point of observation. Vector R denotes thedifference between r’ and r, that is

R =r-r’ (1.21)

Similarly, for a static volume current distribution J (r’), the magnetic vector potentialA(r) is given by

(1.22)

where the following definition for A has been used:

B= XA(1.23)


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For radiation to happen, as has already been pointed out, both charge distribution andcurrent distribution have to be time-varying quantities. Intuitively, we thus modify eqs.

(1.20) and (1.22) to

(1.24 and 1.25)

Finally, for the time-varying case, a finite propagation delay can be expected between thesource and its effect. Assuming v to be the speed of propagation, we thus further modify

the above equations to

(1.26 and 1.27)

These are the so-called ‘retarded’ potentials.

a. Equations for A and V for the time-varying case

Using eq. (1.23) along with the Maxwell’s equation

XE=-( B/ t) (1.28)

we can write

XE=-( ( XA)/ t) (1.29)

That is

X [E+( A/ t)] =0 (1.30)

Hence we define

E+( A/ t)] =- V (1.31)

That is


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E = -( A/ t)] - V (1.32)

Contrast eq. (1.32) with the corresponding definition

E = - V (1.33)

for the static case.

Let us now use the Maxwell’s equation

XH=( D/ t)+J (1.34)

along with eqs. (1.32) and (1.23) to get

2A- ( 2A/ t2)= ( .A)+ ( V/ t)-J (1.35)

Then use

.D= (1.36)

and eq. (1.32) to get

2V + . ( A/ t) = - / (1.37)

Eqs. (1.35) and (1.37) represent the two coupled differential equations for the unknown potentials A and V. It has been shown that the following condition needs to be

additionally satisfied if solutions involving retarded potentials are desired.

( .A) = - ( V/ t) (1.38)

Eq. (1.38) is known as Lorentz gauge condition. Using this equation, eqs. (1.35) and(1.37) become

2A-( 2A/ t2)= -J (1.39)

and

2V-( 2V/ t2)= - / (1.40)


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Eqs. (1.39) and (1.40) represent the two independent equations for the two unknown potentials A and V. It can be shown that the intuitively constructed solutions given by

eqs. (1.26) and (1.27) do indeed satisfy the above two equations (see section 10.10 of

Jordan and Balmain’s book).

b. Potential functions for the sinusoidally time-varying case

In this case, eqs. (1.39) and (1.40) become

2A + k

2A= -J (1.41)

and

2V + k

2V= - / (1.42)

and the solutions to them can be written as

(1.43 and 1.44)

where the phase variation exp (-jR) has been brought in, assuming a uniform planewave.

7. Infinitesimally small dipole (alternating-current element)

An infinitesimally small dipole (also known as the alternating-current element) is shown

in Figure 2. It consists of a time-varying current-element Idl where I represents the

constant current flowing in the dipole and ‘dl’ represents the length of the dipole. ‘dl’ isassumed to be very small compared to the free-space wavelength involved and, hence,

the current I is assumed to be constant throughout the length of the dipole. Also, the

cross-section of the antenna is assumed to be very small compared to the free-space

wavelength involved.

The case of the infinitesimally-small dipole is not practically realizable in most

situations, but can be thought of as a building block for realizing real antennas.


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Figure 2 Infinitesimally small dipole antenna

Since the length of the dipole is assumed to be along the z-axis, the magnetic vector

potential will have only a z-component. An application of eq. (1.43) yields

Az=[/(4)] [Idl {exp(-jr)}/r ] (1.45)

In carrying out the volume integration of the current density, the following logic is used:

The result of integration of J over the dipole’s cross-section is just the current I.

Moreover, since I is constant over ‘dl’, the result of integration along z is simply Idl.

Also note that the antenna is assumed to be centered at the origin of the co-ordinatesystem. This means

r’=0 (1.46)

and

R =r (1.47)

It is also easy to show that

Ar=Azcos(1.48)


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A

=-Azsin(1.49)

A

=0 (1.50)

Since the antenna is symmetrical in the plane, we can also assume that neither Anoranyofthe field components varies with That is,

( A/

and similar equations for the field components.

Once A is known, we can use eq. (1.23) to derive the various components of the magnetic

field H. We get

Hr=0 (1.52)

H

=0 (1.53)

and

H

= [Idlsin

/(4

)][exp(-j

r)][ (j

r)

r

(1.54)

Then we use the Maxwell’s equation

XH=( D/ t)+J (1.55)

to derive expressions for the various components of the lectric field. We get

Er=[2Idlcos/(4 j)][exp(-jr)][ (jr2) r

(1.56)

E

=[Idlsin/(4)][exp(-jr)][ (jr) r

jr3(1.57)

E

= 0 (1.58)

a. Far-field expressions

In the radiated field equations derived above, the (1/r) terms constitute the far-fieldwhereas the complete (1/r)+(1/r

2)+(1/r

3)+… terms constitute the near field.


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Generally, we are concerned with far-fields only. Thus neglecting the (1/r 2) and (1/r

3)

terms, the field expressions for the radiatied field of an infinitesimally small dipole

antenna become

Er=0 (1.59)

E

=H

(1.60)

E

= 0 (1.61)

Hr=0 (1.62)

H

=0 (1.63)

H

= [Idlsin/(4)][exp(-jr)][ (jr)(1.64)

b. Radiated Power

Radiated power can be computed by first finding the average Poynting vector and thenintegrating its radial component Pravg over a spherical surface centered at the element

(see Figure 3).

Figure 3 Speherical surface of integration

Considering only the far fields, it can be shown that

Pravg=(2/2r

2)I

2dl

2sin

2/(16

2) (1.65)

The elemental area will be taken as the area of a strip on the surface of the sphere asshown in Figure 3. That is


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da=2r2sind (1.66)

After integrating the product of Pravg and da over =0 to =, the total far-field radiated power can be shown to be given by

Prad =[802 (dl/)2] Ieff (1.67)

where

Ieff =I/sqrt(2) (1.68)

is the effective current flowing in the antenna.

c. Radiation Resistance

It is customary to define the total radiated power of an antenna in terms of the ‘radiation

resistance’ R rad of the antenna.The radiati on resistance is that hypothetical resistance

which, assuming an rms current I eff flowing through the resistance which is same as

the curr ent f lowing through the antenna, wil l dissipate the same amount of power as

the power r adiated by the antenna.

Mathematically speaking, we can write

Prad=R rad Ieff 2(1.69)

whereby the radiation resistance of the infinitesimally dipole antenna turns out to be

R rad =802 (dl/)

2(1.70)

8. Short antennas (elementary dipole and elementarymonopole)

The elementary dipole antenna has a length large enough so as not to support theconstant-current assumption along the length of the antenna. For cases where the total

length of the dipole is less than quarter-wavelength, it is sufficiently accurate to describethe current distribution along the antenna length as a linear function of z as shown inFigure 4a.

The elementary monopole antenna shown in Figure 4b is similar in construction but ismounted over a ground plane (assumed to be perfectly conducting). For cases where the

total length of the monopole is less than one-eighth wavelength, the current distribution

along the antenna length can be assumed to be linear. For the sake of finding the radiation


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field, the effects of the gorund plane can be incorporated by bringing in an ‘image’ of the physical antenna on the other side of the ground plane as shown in Figure 4b.

Figure 4 Short dipole (a) and short monopole (b)

a. Radiation field of the elementary dipole

Assuming a center-fed case for which the linear current distribution shown in Figure 9.4a

is valid, one can easily show that the z-component of the magnetic vector potential is

given by

Az=[

/(8

)] [Il {exp(-jr)}/r ] (1.71)

Note that one has to integrate I(z)dz from z=-l /2 to z=l /2 to get to this result (see class

notes for more details).

After this, the rest of the analysis runs along lines similar to those followed for the

infinitesimally-small antenna. We can show that, for far fields,

Er=0 (1.72)

E=

H (1.73)

E

= 0 (1.74)

Hr=0 (1.75)

H

=0 (1.76)


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H

= [Il sin/(8)][exp(-jr)][ (jr)(1.77)

b. Radiation resistance of the elementary dipole

The derivation for radiation resistance is also similar to what was done for the

infinitesimally-small dipole. It can be shown that

R rad =202 (l /)

2(1.78)

Note that, compared to the infinitesimally-small dipole, a factor of four has come in, in

the formula for the radiation resistance.

c. Radiation field of the elementary monopole Eqs. (9.50)-(9.55) would still apply as long as we remember that the monopole would

radiate only in the upper hemisphere.

d. Radiation resistance of the elementary monpole

Because the total radiated power in this case is only for the upper half of the sphere, theradiation resistance is given by

R rad =102 (l /)

2(1.79)

9. Dipoles and monopoles of arbitrary length

For dipoles and monopoles of arbitrary length, it is customary to assume a sinusoidal

current distribution as shown in Figure 5, for the center-fed case.

Figure 5 Center-fed dipole and monopole of arbitrary length


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a. Analysis of center-fed dipole of arbitrary length

In this case, we write the current as

I (z)= Im sin {(H-z)} for z>0 (1.80)

I (z)= Im sin {(H+z)} for z


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Az=[m/(4)][exp(-jr)/r][2/{sin2}][cos /2)cos()}]

(1.88)

Using the same methodology that we used for the infinitesimally-small dipole, it can be

shown that, for far fields,

Er=0 (1.89)

E

=H

(1.90)

E

= 0 (1.91)

Hr=0 (1.92)

H

=0 (1.93)

H

= =[jm/(2)][exp(-jr)/r][cos /2)cos}]/[sin

(1.94)

The average Poynting vector can be shown to be

Pravg=[(m

/(8

2

r

2

)][cos

2

/2)cos

sin

2

]

(1.95)

After integrating the product of Pravg and da over =0 to =, the total far-field radiated power can be shown to be given by

Prad ==[(m

/(2)] I(1.96)

where I is a definite integral defined by

I= (1.97)


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Since I can not be found analytically, we need to use numerical techniques. Twodifferent numerical techniques are possible:

1). Use trapezoidal or Simpson’s rule for numerical integration. Using trapezoidal rule(5 degree intervals), the approximate value of I turns out to be 0.609. Please see pp. 330-

331 of Jordan and Balmain’s book for more details.

2). Transform I into an infinite series and then numerically sum that series. This waytoo, the value of I turn out to be approximately equal to 0.609. Please see pp. 330-331 of

Jordan and Balmain’s book for more details.

Thus, the total far-field radiated power of a center-fed half-wave dipole turns out be

approximately given by

Prad ==[(0.609m

/(2)] (1.98)

which, in terms of the rms current Ieff , can be written as

Prad ==[(0.609eff

/()] (1.99)

The radiation resistance of the center-fed half-wave dipole thus can be written as

R rad ==[(0.609/()] (1.100)

which, for free space, works out to approximately 73 ohm .

c. Quarter-wave monopole The analysis of the quarter-wave monopole (after including its image antenna andremoving the ground plane) is similar to that for the half-wave dipole. The only

difference is that the radiation will happen in the upper hemisphere only. Thus

R rad ==[(0.609/(2)] (1.101)

which, for free space, works out to approximately 36.5 ohm .

10. Unsolved problemsAssume free-space as the medium, unless otherwise stated.

1) Solve problems 7,8 and 9 from Jordan and Balmain’s book (pp. 332-333). For problem 9, use the sine and cosine integral definitions given in section 10.08.

2) Find the radiation resistance of a Herzian dipole (i.e., an infinitesimally small

dipole) of length =(/40) where is the free-space wavelength involved.


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3) An antenna whose radiation resistance is 300 ohms operates at a frequency of1GHz and with a current of 3 Amperes. Find the radiated power.

4) The field amplitude due to a half-wave dipole at 10Km is 0.1V/m. The antennaoperates at 100MHz. Find the length of the antenna. Also find the radiated power per m

2.

5) What is the length of a half-wave dipole antenna at 1MHz, if the antenna is

situated in a perfect dielectric medium whose dielectric constant is equal to 10?6) Calculate the radiation resistance of a quarter-wave dipole antenna using a) linear

current distribution, b) sinusoidal current distribution. Compare the two values

thus obtained.

1. ANT_ENGG_CHAPTER1_REV2

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