Antenna Fundamentals

Hon Tat Hui Antennas

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Antenna Fundamentals1 Introduction

Antennas are device designed to radiate electromagnetic energy efficiently in a prescribed manner. It is the current distributions on the antennas that produce the radiation. Usually these current distributions are excited by transmission lines or waveguides.

Transmission line Current distributionsAntenna


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2 Antenna Parameters2.1 Poynting Vector and Power Density

Instantaneous Poynting vector:

2, , , , , ,

Re , , Re , , (W/m )j t j tx y z t x y z t

x y z e x y z e

p E H

E H

Average Poynting vector:

* 2av 1 Re , , , , (W/m )2 x y z x y z P E HTime expressions:E(x,y,z,t)H(x,y,z,t)Phasor expressions:E(x,y,z)H(x,y,z)

Note that Poynting vector is a real vector. Its magnitude gives the instantaneous or average power density of the electromagnetic wave. Its direction gives the direction of the power flow at that particular point.

Note:


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2.2 Power Intensity

2 av W/srU r Psr = steradian, unit for measuring the solid angle.Solid angle is the ratio of that part of a spherical surface area S subtended at the centre of a sphere to the square of the radius of the sphere.

r

S

2 sr

Sr

Sphericalsurface

The solid angle subtended by a whole spherical surface is therefore:

(sr) 44 22

rr

o

Note that U is a function of direction (,) only and not distance (r).


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2.3 Radiated Power

*rad av

1 Re[ ] (W)2s s

P P ds E H ds

Pav

Antenna

nds sin2 ddr

Note that the integration is over a closed surface with the antenna inside and the surface is sufficiently far from the antenna (far field conditions).

r


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Example 1Find the total average radiated power of a Hertzian dipole.

Solution

av2

2 22

2

1 1Re Re2 21 Re2 2

sin (W/m )2 4

E H

EE E

kIdr

r

r r

r

P E H a

a a

a


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rad av

2 2 22

20 0

2

sin sin 2 4

(W)3

s

P

kId r d dr

Id

r r

P ds

a a


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Example 2Find the total average radiated power of a half-wave dipole.

For a half-wave dipole:Solution

cos 2 cos60 , sin

jkr

me EE j I H

r

2

av

222

2

2

15 cos[( / 2)cos ] (W/m )sin

m

E

Ir

r

r

P a

a


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rad av

2222

20 0

22

0

15 cos[( / 2)cos ] sin sin

cos [( / 2)cos ] 30 (W)sin

s

m

m

P

I r d dr

I d

r r

P ds

a a

The above remaining integral can be evaluated numerically to give:

2rad 36.54 (W)mP I


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Hence for a /4 monopole over a ground plane with a maximum current at its base = Im, the radiated power is half that of a /2 dipole, i.e.,

2rad 18.27 (W)mP I

Why?? Think about it!


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2.4 Radiation PatternA radiation pattern (or field pattern) is a graph that describes the relative far field value, E or H, with direction at a fixed distance from the antenna. A field pattern includes an magnitude pattern |E| or |H| and a phase pattern E or H.

A power pattern is a graph that describes the relative(average) radiated power density |Pav| of the far-field with direction at a fixed distance from the antenna.

By the reciprocity theorem, the radiation patterns of an antenna in the transmitting mode is same as the those for the antenna in the receiving mode.


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A radiation pattern shows only the relative values but not the absolute values of the field or power quantity. Hence the values are usually normalized (i.e., divided) by the maximum value.


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For example, the radiation pattern of the Hertzian dipole can be plotted using the following steps.

0sin , 0 2

4fixed

jkr

kId eE j

rr

(1) Far field:

(2) Far field magnitude:

0sin , 0 2

4fixed

kIdE

rr


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(3) Normalization:

n

0sin4 sin , 0 2

fixed4

kId rE kId

rr

(4) Plot plane pattern (fix at a chosen value, for example = 0)

|E|n with at = 0 & 180


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(5) Plot plane pattern (fix at a chosen value, for example = 90)

|E|n with at = 90

See animation Field Behaviour and Radiation Pattern


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2.5 PolarizationThe polarization of an antenna in a given direction is defined as the polarization of the plane wave transmitted by the antenna in that direction. The polarization of a plane wave is the figure the tip of the instantaneous electric-field vector E traces out with time at a fixed observation point. There are three types of typical antenna polarizations: the linear, circular, and ellipticalpolarizations, corresponding to the same three types of typical plane wave polarizations.


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Ex

Ey

Eectric-field vector

Ex

Ey


Ex

Ey


Linearly polarized Circularly polarized Elliptically polarized

See animation Polarization of a Plane Wave - 2D View

See animation Polarization of a Plane Wave - 3D View


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A plane wave is linearly polarized at a fixed observation point if the tip of the electric-field vector at that point moves along the same straight line at every instant of time.

(a) Linear polarization

(b) Circular PolarizationA plane wave is circularly polarized at a a fixed observation point if the tip of the electric-field vector at that point traces out a circle as a function of time.

2.5.1 Polarization of Plane Waves


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A plane wave is elliptically polarized at a a fixed observation point if the tip of the electric-field vector at that point traces out an ellipse as a function of time. Elliptically polarization can be either right-handed or left-handed corresponding to the electric-field vector rotating clockwise (right-handed) or anti-clockwise (left-handed).

(c) Elliptical Polarization

Circular polarization can be either right-handed or left-handed corresponding to the electric-field vector rotating clockwise (right-handed) or anti-clockwise (left-handed).


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The instantaneous expression for E is:

kztEkztE

ejEeEtz

yx

jkztjy

jkztjx

sincos

Re,

00

00

yx

yxE

For example, consider a plane wave:

jkzy

jkzx

yx

ejEeE

EE

00

yx

yxEjkz

yy

jkzxx

ejEE

eEE

0

0

0 0= cos , sinx x y yX E E t kz Y E E t kz

Note that the phase difference between Ex and Ey is 90.

Let:

Ex0 and Ey0 are both real numbers


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Case 1: 0 or 0, thenxo yoE E 0 or 0X Y

Both are straight lines. Hence the wave is linearlypolarized.

Case 2: , thenxo yoE E C 2 2 2 2 2 2cos sinX Y C t kz t kz C

X and Y describe a circle. Hence the wave iscircularly polarized.

Case 3: , thenxo yoE E

2 2

2 22 20 0

cos sin 1x y

X Y t kz t kzE E

X and Y describe an ellipse. Hence the wave iselliptically polarized.


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2.5.2 Axial RatioThe polarization state of an EM wave can also be indicated by another two parameters: Axial Ratio (AR) and the tilt angle (). AR is a common measure for antenna polarization. It definition is:

OAAR , 1 AR , or 0 dB AR dBOB

where OA and OB are the major and minor axes of the polarization ellipse, respectively. The tilt angle is the angle subtended by the major axis of the polarization ellipse and the horizontal axis.


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= tilt angle0 180


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AR = 1, circular polarization1 < AR < , elliptical polarizationAR = , linear polarizationAR can be measured experimentally!

For example:

Very often, we use the AR bandwidth and the AR beamwidth to characterize the polarization of an antenna. The AR bandwidth is the frequency bandwidth in which the AR of an antenna changes less than 3 dB from its minimum value. The AR beamwidth is the angle span over which the AR of an antenna changes less than 3 dB from its mimumum value.


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AR at

Radiation patternwith a rotating linear source

3 dB AR beamwidth

Test antenna(receiving) Fast-rotating dipole

antenna (transmitting)


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Frequency

Axial ratio (dB)

3dB

AR bandwidth


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2.6 Input ImpedanceThe input impedance ZA of a transmitting antenna is the ratio of the voltage to current at the terminals of the antenna.

A A AZ R jX RA = input resistanceXA = input reactance

A r LR R R Rr = radiation resistanceRL = loss resistanceIf we know the input impedance of a transmitting antenna, the antenna can be viewed as an equivalent circuit.


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whereg g gZ R jX

Zg = internal impedance of the excitation sourceRg = internal resistance of the excitation sourceXg = internal reactance of the excitation source

VgZg

Equivalent circuit

Vg

Rg

Xg

Rr

XA

RLab

a

b

Ig

Ig

Ig = antennaterminalcurrent

Excitation source

Transmitting antenna


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The knowledge of ZA is required when connecting an antenna to its driving circuit.

The radiation resistance Rr can be calculated from the power radiated Prad as:

*If , antenna is matched.A gZ ZIf ,A gZ Z antenna is not matched and a

matching circuit is required.

2rad

12 g r

P I R

2loss

12 g L

P I R

Power loss as heat in the antenna:


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Power loss in the internal resistance of the excitation source:2

internal12 g g

P I R

Maximum power transfer from the excitation source to the antenna occurs if the antenna is matched. That is,

*A gZ Z, r L g A gR R R X X

If the antenna is connected to the driving circuit via a transmission line with a characteristic impedance Z0, then the antenna should be matched to the characteristic impedance of transmission line. That is,

0 0, , 0A r L AZ Z R R Z X


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The impedance looking into the terminals of a receiving antenna is called internal impedance Zin. In general, Zin ZA. However, when the antenna size is small compared to the wavelength, Zin ZA. For dipole antennas, Zin ZA when dipole length . The internal impedance is used to model the equivalent circuit of a receiving antenna as the input impedance is used to model the equivalent circuit of a transmitting antenna (see later).Students who want to know more on this topic can read the following article:C. C. Su, On the equivalent generator voltage and generator internal impedance for receiving antennas, IEEE Transactions on Antennas and Propagation, vol. 51(2), pp. 279-285, 2003.


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Example 3Calculate the radiation resistance of a Hertzian dipole.SolutionFrom example 1, the radiated power Prad of a Hertzian dipole is: 2

rad 3IdP

Therefore,

22

rad

22

1 2 3

80

r

r

IdP I R

dR


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Example 4Calculate the radiation resistance of a half-wave dipole.SolutionFrom example 2, the radiated power Prad of a half-wave dipole is:

2rad 36.54 mP I

Therefore,2 2

rad1 36.542

73.1

m r m

r

P I R I

R

This result is based on the assumption of an infinitely thin dipole (wire diameter 0). For a finite thickness dipole, the radiation resistance is generally greater than this value.


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Note that the input reactance XA of an antenna cannot be found from the radiated power. It can be calculated by other methods such as Moment Method or the Induced EMFmethod. For an infinitely think half-wave dipole,

XA = 42.5 For an infinitely thin quarter-wave monopole over a large ground plane,

XA = 21.3

Students who want to know more on this can read the following book:John D. Kraus, Antennas, McGraw-Hill, New York, 1988, Chapters 9 & 10.


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2.7 Reflection Coefficient

0

0

(dimensionless)AA

Z ZZ Z

The reflection coefficient of a transmitting antenna is defined by:

can be calculated (as above) or measured. The magnitude of is from 0 to 1. When the transmitting antenna is not macth, i.e., ZA Z0, there is a loss due to reflection (return loss) of the wave at the antenna terminals. When expressed in dB, is always a negative number. Sometimes we use S11 to represent .


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2.8 Return LossThe return loss of a transmitting antenna is defined by:

return loss 20log (dB)

Possible values of return loss are from 0 dB to dB. Return loss is always a positive number.


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2.9 VSWRThe voltage standing wave ratio (VSWR) of a transmitting antenna is defined by:

1VSWR (dimensionless)1

Same as and the return loss, VSWR is also a common parameter used to characterize the matching property of a transmitting antenna. Possible values of VSWR are from 1 to . VSWR=1 perfectly matched. VSWR = completely unmatched.


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2.10 Impedance Bandwidth

Frequency

|| or |S11| (dB)

-10dB

Impedance bandwidth

fL fUfC


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Impedance bandwith 100%U LC

f ff

Note that when || = -10 dB,

1 1 0.3162VSWR =1 1 0.3162

=1.93 2

Hence the impedance bandwidth can also be specified by the frequency range within which VSWR 2.


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2.11 DirectivityThe directivity D of an antenna is the ratio of the radiation intensity U in a given direction (, ) to the radiation intensity averaged over all directions U0.

0 rad rad

, , 4 ,,/ 4

U U UDU P P

Maximum directivity D0 is the directivity in the maximum radiation direction (0, 0).

max max0

0

4rad

U UDU P


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2.12 GainThe gain or power gain of an antenna in a certain direction (, ) is defined as:

in

4 ,radiation intensity,total input power / 4

UGP

where Pin is the input power to the antenna and is related to the radiated power Prad as:

in radP P


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Taking the efficiency into account, the gain and the directivity are related by:

, ,G D Similar to the maximum directivity, a maximum gainG0 can be defined and which is related to the maximum directivity D0 by:

max0 0

in

4 UG DP

Here is the efficiency of the antenna. It accounts for the various losses in the antenna, such as the reflection loss, dielectric loss, conduction loss, and polarization mismatch loss.


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Example 5

Find the maximum gain and directivity of a Hertzian dipole. Assume that the antenna is lossless with an efficiency equal to 1.Solution

2 2

av 2sin

2 4kId

r

rP a

2

2 2av sin2 4

kIdU r

P

2

rad 3IdP


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2

2

22

rad

4 sin4 , 32 4, sin2

3

kIdUDP Id

231 , , sin2

G D

0 090 90

3 1.52

G D


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Example 6

Find the maximum gain and directivity of a half-wave dipole. Assume that the antenna is lossless with an efficiency equal to 1.Solution

22av 215 cos[( / 2)cos ]sinmIr rP a2

rad 36.54 (W)mP I

222 av 15 cos[( / 2)cos ]sinmIU r P


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22

2rad

2

15 cos[( / 2)cos ]44 , sin,36.54

cos[( / 2)cos ] 1.64sin

m

m

IUDP I

2cos[( / 2)cos ]1 , , 1.64 sinG D 0 090 90 1.64G D


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2.13 Effective Area

The effective aperture (area) of a receiving antennalooking from a certain direction (,) is the ratio of the average power PL delivered to a matched load to the magnitude of the average power density Pavi of the incident electromagnetic wave at the position of the antenna multiplied by the normalized power pattern |Pav(,)| of that antenna.

av

,,

Le

avi

PAP

P


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A maximum effective area Aem can be defined when the antenna is receiving in its maximum-directivity direction. That is,

2

em 04A D

The effective area is related to the directivity as (see Supplementary Notes):

2

, ,4e

A D


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2.14 Open Circuit VoltageA receiving antenna can be modelled as an equivalent circuit as follows:

ZL Equivalent circuit

VocRL

XLRin

Xin

ab

a

b

IL

IL

Receiving antenna

Incident wave

ZL = RL + jXL = load impedance Zin = Rin + jXin = internal impedance

a is positive with respect to b


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The open-circuit voltage Voc is defined as the voltage which appears at the terminals of a receiving antenna when the antenna is excited by an incident wave and the terminals are left open.

1oc i

m

V dI

I E

where current on the antenna when the antenna is excited at the terminal

current at the terminalincident electricfield

length of the wire antenna

m

i

distribution

I

I

E

In order to produce a positive Voc, I and Ei must be in opposite senses.


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Reciprocity Theorem

V1

I1

dI2

dV2

Case 1 Case 2

1 2

1 2

I dIV dV

Im

Circuit Voltage Expression-Proof of the Open


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Putting

1 2, m A iV I Z dV E d

we have,

112 2

1

2 11

i

m A

im A

I E dIdI dVV I Z

I I E dI Z

In vector form,

2 11

im A

I dI Z

I E


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Putting I1 equal to I and noting that I2 is the short-circuit current at the terminal of the antenna, by Theveninstheorem, the open-circuit voltage Voc at the antenna terminal can then be expressed as:

21

oc A im

V I Z dI

I E

(For a more detailed explanation on the reciprocity theorem, see Chapter 11, ref. [4].)


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References:

1. David K. Cheng, Field and Wave Electromagnetic, Addison-Wesley Pub. Co., New York, 1989.

2. John D. Kraus, Antennas, McGraw-Hill, New York, 1988.3. C. A. Balanis, Antenna Theory, Analysis and Design, John Wiley

& Sons, Inc., New Jersey, 2005.4. E. C. Jordan, Electromagnetic Waves and Radiating Systems,

Prentice-Hall, ley, New York, 1998.5. Fawwaz T. Ulaby, Applied Electromagnetics, Prentice-Hall Inc.,

Englewood Cliffs, N. J., 1968.6. Joseph A. Edminister, Schaums Outline of Theory and Problems

of Electromagnetics, McGraw-Hill, Singapore, 1993.7. Yung-kuo Lim (Editor), Problems and solutions on

electromagnetism, World Scientific, Singapore, 1993.

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Antenna Fundamentals