Antennas

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Antenna Fundamentals 1 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 Hon Tat Hui

Antenna 1

Current distributions Antennas

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Antenna Parameters

2.1 Poynting Vector and Power Density Instantaneous Poynting vector: p  E  x, y , z , t   H  x, y , z , t   Re E  x, y, z  e jt   Re H  x, y, z  e jt  (W/m 2 ) Note: Time expressions: E(x,y,z,t) H(x,y,z,t) 2 Phasor expressions: E(x,y,z) H(x,y,z)

Average Poynting vector: 1 * Pav  Re E  x, y, z   H  x, y, z  (W/m ) 2 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.



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

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2.2 Power Intensity U  r Pav 2

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

 W/sr 

sr = 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. S Spherical S   2  sr  surface r  The solid angle subtended o by a whole spherical r surface is therefore: 4r 2 

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r 3

2

 4 (sr)

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2.3 Radiated Power

1 * Re[ ]  ds Prad   P  ds  E  H s av   2 s

(W)

ds  r 2 sin dd nˆ  Pav  Antenna

r

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). Hon Tat Hui

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Example 1 Find the total average radiated power of a Hertzian dipole. Solution 1 1  Pav  ReE  H   Re  E  H  ar 2 2 2



E 1  E E   Re  ar  ar  2    2 2

 kId  sin 2  2  a (W/m ) r 2 2 4 r Hon Tat Hui

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Prad    Pav  ds s

 kId   2 4

2  2

 Id   3 

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sin 2  2 a  r 0 0 r 2 r sin  d d ar

2

(W)

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Example 2 Find the total average radiated power of a half-wave dipole. Solution For a half-wave dipole:  e  jkr  cos  2  cos  Eθ , H  Eθ  j 60 I m    sin   r  2

E Pav  ar 2 15 I  r Hon Tat Hui

2 m 2





2

cos[( / 2)cos ] ar (W/m 2 ) sin  7

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Prad    Pav  ds s

15I  r

2  2 m 2 0 0







2

cos[( / 2)cos ] ar  r 2 sin  d d ar sin 



2 cos [( / 2)cos ] 2  30 I m  d sin  0

(W)

The above remaining integral can be evaluated numerically to give: Prad  36.54 I m2 Hon Tat Hui

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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.,

Prad  18.27 I m2

(W)

Why?? Think about it!

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2.4 Radiation Pattern A 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. Hon Tat Hui

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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. (1) Far field:

 kId   e  Eθ  j sin θ ,   4  r   jkr

0      0    2 r fixed 

(2) Far field magnitude:

 kId  Eθ  sin θ , 4 r Hon Tat Hui

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0      0    2 r fixed  Antennas

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(3) Normalization:  kId  sin θ Eθ n  4 r  sin θ ,  kId  4 r (4) Plot –plane pattern (fix  example  = 0°)

0      0    2 r fixed  at a chosen value, for

|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 Polarization The 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 elliptical polarizations, corresponding to the same three types of typical plane wave polarizations.

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EE4101 Ey

Ey

Ey

Ex Eectric-field vector

Linearly polarized

Ex

Ex Eectric-field vector

Eectric-field vector

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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2.5.1 Polarization of Plane Waves (a) Linear polarization 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. (b) Circular Polarization

A 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. Hon Tat Hui

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Circular polarization can be either right-handed or left-handed corresponding to the electric-field vector rotating clockwise (right-handed) or anticlockwise (left-handed). (c) Elliptical Polarization 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). Hon Tat Hui

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For example, consider a plane wave: Ex  Ex 0e

E  xˆ E x  yˆ E y  xˆ E x 0 e

 jkz

 yˆ jE y 0 e

 jkz

Ex0 and Ey0 are both real numbers

 jkz

E y   jE y 0 e  jkz

Note that the phase difference between Ex and Ey is 90º. The instantaneous expression for E is:



E z , t   Re xˆ E x 0 e jt  jkz  yˆ jE y 0 e jt  jkz



 xˆ E x 0 cost  kz   yˆ E y 0 sin t  kz 

Let:

X  Ex =Ex 0 cos t  kz  , Y  E y  E y 0 sin t  kz  Hon Tat Hui

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Case 1: Exo  0 or E yo  0, then X  0 or Y  0 Both are straight lines. Hence the wave is linearly polarized. Case 2: Exo  E yo  C , then X 2  Y 2  C 2 cos 2 t  kz   sin 2 t  kz    C 2 X and Y describe a circle. Hence the wave is circularly polarized. Case 3: Exo  E yo , then 2 2 X Y 2 2 t kz   cos    sin   t  kz   1 2 2 Ex 0 E y 0 X and Y describe an ellipse. Hence the wave is elliptically polarized.

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2.5.2 Axial Ratio

The 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: OA AR  , OB

1  AR  , or 0 dB  AR  dB

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. Hon Tat Hui

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 = tilt angle 0 ≤  ≤ 180º



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For example: AR = 1,  circular polarization 1 < AR < ,  elliptical polarization AR = ,  linear polarization AR can be measured experimentally! 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. Hon Tat Hui

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3 dB AR beamwidth

Radiation pattern with a rotating linear source



AR at 

Test antenna (receiving) Hon Tat Hui

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Fast-rotating dipole antenna (transmitting) Antennas

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Axial ratio (dB)

3dB

AR bandwidth Hon Tat Hui

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Frequency Antennas

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2.6 Input Impedance The input impedance ZA of a transmitting antenna is the ratio of the voltage to current at the terminals of the antenna. Z  R  jX A

A

A

RA = input resistance XA = input reactance RA  Rr  RL

Rr = radiation resistance RL = loss resistance If we know the input impedance of a transmitting antenna, the antenna can be viewed as an equivalent circuit. Hon Tat Hui

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Excitation source

Vg Zg

Ig 

Ig 

a b

 Equivalent

circuit

a

Vg

Rr

Rg

RL

Xg

XA

Transmitting antenna

b

where Z g  Rg  jX g

Ig = antenna terminal current

Zg = internal impedance of the excitation source Rg = internal resistance of the excitation source Xg = internal reactance of the excitation source Hon Tat Hui

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The knowledge of ZA is required when connecting an antenna to its driving circuit. * If Z A  Z g , antenna is matched. If Z A  Z g ,

antenna is not matched and a matching circuit is required.

The radiation resistance Rr can be calculated from the power radiated Prad as: 1 2 Prad  I g Rr 2 Power loss as heat in the antenna: 1 2 Ploss  I g RL 2 Hon Tat Hui

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Power loss in the internal resistance of the excitation source: 1 2 Pinternal  I g Rg 2 Maximum power transfer from the excitation source to the antenna occurs if the antenna is matched. That is, Z A  Z g* Rr  RL  Rg , X A   X g 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, Z A  Z 0 , Rr  RL  Z 0 , X A  0 Hon Tat Hui

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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. Hon Tat Hui

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Example 3 Calculate the radiation resistance of a Hertzian dipole. Solution From example 1, the radiated power Prad of a Hertzian dipole is: 2  Id  Prad  3 

Therefore,

1 2  Id  Prad  I Rr  2 3 

2

2

 Hon Tat Hui

d   Rr  80    Ω    2

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Example 4 Calculate the radiation resistance of a half-wave dipole. Solution From example 2, the radiated power Prad of a half-wave dipole is: Prad  36.54 I m2 Therefore, 1 2 Prad  I m Rr  36.54 I m2 2  Rr  73.1 Ω

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. Hon Tat Hui

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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 EMF method. 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. Hon Tat Hui

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2.7 Reflection Coefficient

The reflection coefficient of a transmitting antenna is defined by: Z A  Z0  Z A  Z0

(dimensionless)

 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 . Hon Tat Hui

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2.8 Return Loss

The 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 VSWR The voltage standing wave ratio (VSWR) of a transmitting antenna is defined by: 1  VSWR  1 

(dimensionless)

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. Hon Tat Hui

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2.10 Impedance Bandwidth

|| or |S11| (dB)

-10dB

fL

fC

fU

Impedance bandwidth Hon Tat Hui

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Frequency Antennas

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fU  f L Impedance bandwith   100% fC

Note that when || = -10 dB, 1   1  0.3162 VSWR  = 1   1  0.3162 =1.93 2

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

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2.11 Directivity The 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. U  ,  U  ,  4 U  ,  D  ,     U0 Prad / 4 Prad

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

U max 4 U max  D0  U0 Prad Hon Tat Hui

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2.12 Gain The gain or power gain of an antenna in a certain direction (, ) is defined as: 4 U  ,  radiation intensity G  ,    total input power / 4 Pin

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

Pin  Prad Hon Tat Hui

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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. Taking the efficiency into account, the gain and the directivity are related by: G  ,    D  ,  Similar to the maximum directivity, a maximum gain G0 can be defined and which is related to the maximum directivity D0 by: 4 U max G0    D0 Pin Hon Tat Hui

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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

 kId  sin 2  Pav  ar 2 2 4 r

 Id  Prad  3 

2

2

 kId  2 U    r Pav  sin  2 4 2

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2

 kId  4 sin 2  4 U  ,  3 2  2 4 D  ,     sin  2 Prad 2  Id  3   1



3 2 G  ,   D  ,   sin  2

G0  90  D0  90

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3   1.5 2

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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





2

15 I m2 cos[( / 2)cos ] Pav  ar 2 r sin  Prad  36.54 I m2 U    r Pav  2

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15 I



2 m

45

(W)





cos[( / 2)cos ] sin 

2

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4 U  ,  D  ,    Prad

4

15 I





2 m





cos[( / 2)cos ] sin  36.54 I m2



cos[( / 2)cos ]  1.64 sin 

 1



2



2



cos[( / 2)cos ] G  ,   D  ,   1.64 sin 

2

G0  90  D0  90  1.64 Hon Tat Hui

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2.13 Effective Area

The effective aperture (area) of a receiving antenna looking 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. PL Ae  ,   Pavi Pav  ,  Hon Tat Hui

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

2 Ae  ,   D  ,  4 A maximum effective area Aem can be defined when the antenna is receiving in its maximum-directivity direction. That is,

2 Aem  D0 4 Hon Tat Hui

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2.14 Open Circuit Voltage A receiving antenna can be modelled as an equivalent circuit as follows: IL

Incident wave

IL ZL

a b

 Equivalent circuit

a Voc 

RL

Rin

XL Xin b

Receiving antenna a is positive with respect to b Hon Tat Hui

ZL = RL + jXL = load impedance Zin = Rin + jXin = internal impedance 49

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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. In order to produce a

where

1 Voc    I  Ei d  Im 

positive Voc, I and Ei must be in opposite senses.

I  current distribution on the antenna

when the antenna is excited at the terminal I m  current at the terminal Ei  incident electricfield   length of the wire antenna Hon Tat Hui

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Proof of the Open-Circuit Voltage Expression Reciprocity Theorem Im



I1

V1 

dV2

dI2

Case 2

Case 1

I1 dI 2  V1 dV2 Hon Tat Hui

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Putting V1   I m Z A , dV2  Ei d 

we have,

I1    Ei    d  I1 dI 2  dV2   V1 ImZ A 1 I2   I1    Ei    d   ImZ A 

In vector form, 1 I2   I1  Ei d   ImZ A  Hon Tat Hui

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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 Thevenin’s theorem, the open-circuit voltage Voc at the antenna terminal can then be expressed as: 1 Voc  I 2 Z A    I  Ei d  Im 

(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, AddisonWesley 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, Schaum’s 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. Hon Tat Hui

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