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Antenna Fundamentals

Antenna Fundamentals 4
NUS/ECE EE5308 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 Current distributions Antenna Hon Tat Hui Antenna Fundamentals 1NUS/ECE EE5308 2 Antenna Parameters 2.1 Poynting Vector and Power Density Instantaneous Poynting vector: pE xy,,z,tH xy, ,z,t  jt jt 2  ReEH xy , ,z e Re xy , ,z e (W/m )    Time expressions: Note: Average Poynting vector: E(x,y,z,t) H(x,y,z,t) 1 2 Phasor expressions: PE Re xy ,,zH x,, y z (W/m )    av E(x,y,z) 2 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. Hon Tat Hui Antenna Fundamentals 2NUS/ECE EE5308 2.2 Power Intensity Note that U is a function of direction (θ,) only and 2 not distance (r). Ur P W/sr  av 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  sr  surface 2 r  The solid angle subtended o by a whole spherical r surface is therefore: 2 4r  4 (sr) 2 r Hon Tat Hui Antenna Fundamentals 3NUS/ECE EE5308 2.3 Radiated Power 1 P Pds ReEH ds (W) rad av    2 ss 2 ˆ ds r sindd n  P av  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 Antenna Fundamentals 4NUS/ECE EE5308 Example 1 Find the total average radiated power of a Hertzian dipole. Solution 11  PE  ReH ReEHa   av  r 22 2   E 1 EE    Re aa  rr 22   2 2  kId sin 2  a (W/m ) r 2 24 r Hon Tat Hui Antenna Fundamentals 5NUS/ECE EE5308 P Pds rad av  s 2  2 2  kId sin 2 aa rd sin d rr 2  24 r 00 2  Id  (W) 3 Hon Tat Hui Antenna Fundamentals 6NUS/ECE EE5308 Example 2 Find the total average radiated power of a halfwave dipole. Solution For a halfwave dipole:  jkr cos 2 cos eE θ Ej 60I , H θ m  r sin  2 E  Pa av r 2 2 2 15I cos( / 2)cos 2 m  a (W/m ) r 2   r sin Hon Tat Hui Antenna Fundamentals 7NUS/ECE EE5308 P Pds rad av   s 2  2 2 15I cos( / 2)cos 2 m aa rd sin d rr 2   r sin 00  2 cos ( / 2)cos 2  30Id (W) m  sin 0 The above remaining integral can be evaluated numerically to give: 2 PI 36.54 (W) rad m Hon Tat Hui Antenna Fundamentals 8NUS/ECE EE5308 Hence for a /4 monopole over a ground plane with a maximum current at its base = I , the radiated power is m half that of a /2 dipole, i.e., 2 PI 18.27 (W) rad m Why Think about it Hon Tat Hui Antenna Fundamentals 9NUS/ECE EE5308 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 P of the farfield av 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 Antenna Fundamentals 10NUS/ECE EE5308 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. Hon Tat Hui Antenna Fundamentals 11NUS/ECE EE5308 Hon Tat Hui Antenna Fundamentals 12NUS/ECE EE5308 For example, the radiation pattern of the Hertzian dipole can be plotted using the following steps. (1) Far field: 0   jkr kId e  Ej sinθ, 0 2  θ  4 r   r fixed  (2) Far field magnitude: 0  kId  E sinθ , 0 2  θ 4 r  r fixed  Hon Tat Hui Antenna Fundamentals 13NUS/ECE EE5308 (3) Normalization: kId 0  sinθ  4 r E sinθ , 0 2  θ n kId  r fixed  4 r (4) Plot –plane pattern (fix  at a chosen value, for example  = 0°) E with  at = 0° 180°  n Hon Tat Hui Antenna Fundamentals 14NUS/ECE EE5308 (5) Plot –plane pattern (fix  at a chosen value, for example  = 90°) E with  at = 90°  n See animation “Field Behaviour and Radiation Pattern” Hon Tat Hui Antenna Fundamentals 15NUS/ECE EE5308 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 electricfield 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. Hon Tat Hui Antenna Fundamentals 16NUS/ECE EE5308 E E E y y y E E E x x x Eectricfield vector Eectricfield vector Eectricfield vector Linearly polarized Circularly polarized Elliptically polarized See animation “Polarization of a Plane Wave 2D View” See animation “Polarization of a Plane Wave 3D View” Hon Tat Hui Antenna Fundamentals 17NUS/ECE EE5308 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 electricfield 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 electricfield vector at that point traces out a circle as a function of time. Hon Tat Hui Antenna Fundamentals 18NUS/ECE EE5308 Circular polarization can be either righthanded or lefthanded corresponding to the electricfield vector rotating clockwise (righthanded) or anti clockwise (lefthanded). (c) Elliptical Polarization A plane wave is elliptically polarized at a a fixed observation point if the tip of the electricfield vector at that point traces out an ellipse as a function of time. Elliptically polarization can be either righthanded or lefthanded corresponding to the electricfield vector rotating clockwise (righthanded) or anticlockwise (lefthanded). Hon Tat Hui Antenna Fundamentals 19NUS/ECE EE5308 E and E are x0 y0 For example, consider a plane wave: both real numbers  jkz E E e ˆ ˆ E xE yE x x0 x y  jkz  jkz jkz E jE e ˆ ˆ  xE e yjE e y y0 x0 y0 Note that the phase difference between E and E is 90º. x y The instantaneous expression for E is: j t jkz j t jkz  ˆ ˆ E z,t RexE e yjE e x0 y0 ˆ ˆ  xE cost kz yE sint kz x0 y0 Let: X E =c E os tkz , YEE sin tk z  xx00 y y Hon Tat Hui Antenna Fundamentals 20NUS/ECE EE5308 Case 1: EE 0 or 0, then xo yo XY  0 or 0 Both are straight lines. Hence the wave is linearly polarized. Case 2: EEC, then xo yo 22 2 2 2 2  XYC cos tkz sin tkz C   X and Y describe a circle. Hence the wave is circularly polarized. Case 3: EE , then xo yo 22 XY 22  cos tk z sin tkz1   22 EE xy 00 X and Y describe an ellipse. Hence the wave is elliptically polarized. Hon Tat Hui Antenna Fundamentals 21NUS/ECE EE5308 Example 3 A cross dipole is constructed by a vertical Hertzian dipole along the z axis and a horizontal Hertzian dipole along the x axis. If the horizontal Hertzian dipole is excited with a phase difference of 90 with respect to the vertical Hertzian dipole, what is the polarization of the total farzone field in the direction of  =90,  = 90 z Hertzian dipoles 0 y x  Hon Tat Hui Antenna Fundamentals 22NUS/ECE EE5308 Solutions z Vertical Hertzian  dipole y x  0  0    jk r 0   kId e  jk r  00 0  Far field E j sinθ  Ae sinθ 1   4 r   0    0   kId 00 where Aj 4 r Hon Tat Hui Antenna Fundamentals 23NUS/ECE EE5308 z Horizontal Hertzian  dipole 0 y x    0  0   jk r 0   kId e  jk r  00 0 Far field E jA cos cos e cos cos  2   4π r   jk r  0 Ae sin    jk r  0   kId e 00  j sin  4π r    Hon Tat Hui Antenna Fundamentals 24NUS/ECE EE5308 If the horizontal dipole is excited with a phase difference of 90, then: 00     jk r j 2 jk r    00 E Ae cosc os e  jAe coscos 2     jk r jk r 00 Ae sin jAe sin       The farfield E of the cross dipole antenna is then: Hon Tat Hui Antenna Fundamentals 25NUS/ECE EE5308 00     jk r jk r    00 EE E Ae sinθ  jAe cos cos 12     jk r 0 0s jAein       0   jk r  0  Ae sinθ j cos cos  jsin   At the direction of  =90,  = 90, the farfield is: 0   jk r jk r  00 Ea  Ae 1 Ae  ja    j   Hon Tat Hui Antenna Fundamentals 26NUS/ECE EE5308 In time domain, jt EE Re e  jtjkr   Re Ae aa j     At cos kraa At coskr 2     EEaa    where, EA cost kr   EA cost kr 2   Hon Tat Hui Antenna Fundamentals 27NUS/ECE EE5308 As, 22 22 EE At cos kr At cos kr 2    22  Acos t krAsin t kr    2  A constant The farfield of a crossdipole with a phase difference of 90 in the excitations of the two dipole is circularly polarized. Hon Tat Hui Antenna Fundamentals 28NUS/ECE EE5308 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 , 1 AR, or 0 dB ARdB OB 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 Antenna Fundamentals 29NUS/ECE EE5308  = tilt angle 0≤≤ 180º  Hon Tat Hui Antenna Fundamentals 30NUS/ECE EE5308 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 Antenna Fundamentals 31NUS/ECE EE5308 3 dB AR beamwidth Radiation pattern with a rotating linear source  AR at  in dB scale Test antenna Fastrotating dipole (receiving) antenna (transmitting) Hon Tat Hui Antenna Fundamentals 32NUS/ECE EE5308 Axial ratio (dB) 3dB Frequency AR bandwidth Hon Tat Hui Antenna Fundamentals 33NUS/ECE EE5308 2.6 Input Impedance The input impedance Z of a transmitting antenna is the A ratio of the voltage to current at the terminals of the antenna. ZRjX A AA R = input resistance A X = input reactance A RRR A rL R = radiation resistance r R = loss resistance L If we know the input impedance of a transmitting antenna, the antenna can be viewed as an equivalent circuit. Hon Tat Hui Antenna Fundamentals 34NUS/ECE EE5308 Excitation source I g a  V R g r I g V g  a Equivalent R g R  L b Z g circuit X X g A Transmitting antenna b where I = antenna g terminal ZRjX gg g current Z = internal impedance of the excitation source g R = internal resistance of the excitation source g X = internal reactance of the excitation source g Hon Tat Hui Antenna Fundamentals 35NUS/ECE EE5308 The knowledge of Z is required when connecting an A antenna to its driving circuit. If ZZ , antenna is matched. Ag antenna is not matched and a If ZZ , Ag matching circuit is required. The radiation resistance R can be calculated from the r power radiated P as: rad 2 1 PI R rad g r 2 Power loss as heat in the antenna: 2 1 PI R loss g L 2 Hon Tat Hui Antenna Fundamentals 36NUS/ECE EE5308 Power loss in the internal resistance of the excitation source: 2 1 PI R internal gg 2 Maximum power transfer from the excitation source to the antenna occurs if the antenna is matched. That is, ZZ Ag RR R , XX rL g A g If the antenna is connected to the driving circuit via a transmission line with a characteristic impedance Z , then the 0 antenna should be matched to the characteristic impedance of transmission line. That is, ZZ , RR Z , X  0 A00 rL A Hon Tat Hui Antenna Fundamentals 37NUS/ECE EE5308 The impedance looking into the terminals of a receiving antenna is called internal impedance Z . In general, Z in in Z . However, when the antenna size is small compared to A the wavelength, Z Z . For dipole antennas, Z Z when in A in A 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. 279285, 2003. Hon Tat Hui Antenna Fundamentals 38NUS/ECE EE5308 Example 4 Calculate the radiation resistance of a Hertzian dipole. Solution From example 1, the radiated power P of a Hertzian dipole rad is: 2  Id P rad 3 Therefore, 2 1  Id 2 PIR rad r 23 2 d  2  R 80Ω   r   Hon Tat Hui Antenna Fundamentals 39NUS/ECE EE5308 Example 5 Calculate the radiation resistance of a halfwave dipole. Solution From example 2, the radiated power P of a halfwave rad dipole is: 2 PI 36.54 rad m Therefore, 1 2 2 PIR 36.54I rad mr m 2  R 73.1 Ω 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. Hon Tat Hui Antenna Fundamentals 40NUS/ECE EE5308 Note that the input reactance X of an antenna cannot be A 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 halfwave dipole, X = 42.5  A For an infinitely thin quarterwave monopole over a large ground plane, X = 21.3  A Students who want to know more on this can read the following book: John D. Kraus, Antennas, McGrawHill, New York, 1988, Chapters 9 10. Hon Tat Hui Antenna Fundamentals 41NUS/ECE EE5308 2.7 Reflection Coefficient The reflection coefficient of a transmitting antenna is defined by: ZZ A 0  (dimensionless) ZZ A 0  can be calculated (as above) or measured. The magnitude of  is from 0 to 1. When the transmitting antenna is not macth, i.e., Z≠ Z , there is a loss due to A 0 reflection (return loss) of the wave at the antenna terminals. When expressed in dB,  is always a negative number. Sometimes we use S to represent . 11 Hon Tat Hui Antenna Fundamentals 42NUS/ECE EE5308 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. Hon Tat Hui Antenna Fundamentals 43NUS/ECE EE5308 2.9 VSWR The voltage standing wave ratio (VSWR) of a transmitting antenna is defined by: 1 VSWR (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. Hon Tat Hui Antenna Fundamentals 44NUS/ECE EE5308 2.10 Impedance Bandwidth  or S (dB) 11 10dB f f L f C U Frequency Impedance bandwidth Hon Tat Hui Antenna Fundamentals 45NUS/ECE EE5308 f f UL Impedance bandwith100 f C 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 Antenna Fundamentals 46NUS/ECE EE5308 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 U . 0 UU,, 4U,   D  ,   UP /4 P 0rad rad Maximum directivity D is the directivity in the 0 maximum radiation direction ( ,  ). 0 0 UU 4 max max D 0 UP 0 rad Hon Tat Hui Antenna Fundamentals 47NUS/ECE EE5308 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 P in where P is the input power to the antenna and is in related to the radiated power P as: rad PP in rad Hon Tat Hui Antenna Fundamentals 48NUS/ECE EE5308 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: GD, ,    Similar to the maximum directivity, a maximum gain G can be defined and which is related to the 0 maximum directivity D by: 0 4U max GD  00 P in Hon Tat Hui Antenna Fundamentals 49NUS/ECE EE5308 Example 6 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  kId sin Pa av r 2 24 r 2  Id P rad 3 2  kId 22 Ur P sin  av 24 Hon Tat Hui Antenna Fundamentals 50NUS/ECE EE5308 2  kId 2 4sin 4,  U   3 24 2 D,s  in  2 P 2  Id rad 3 3 2  1 GD  ,   ,  sin    2 3 GD  1.5 00   90  90 2 Hon Tat Hui Antenna Fundamentals 51NUS/ECE EE5308 Example 7 Find the maximum gain and directivity of a halfwave dipole. Assume that the antenna is lossless with an efficiency  equal to 1. Solution 2 2 15I cos( / 2)cos m Pa av r 2   r sin 2 PI 36.54 (W) rad m 2 2 15I cos( / 2)cos 2 m Ur P  av   sin Hon Tat Hui Antenna Fundamentals 52NUS/ECE EE5308 2 2 15I cos( / 2)cos m 4  4,  U    sin D  ,   2 PI 36.54 rad m 2 cos( / 2)cos  1.64  sin 2 cos( / 2)cos  1 GD  ,   ,  1.64    sin GD 1.64 00 90 90 Hon Tat Hui Antenna Fundamentals 53NUS/ECE EE5308 2.13 Effective Area The effective aperture (area) of a receiving antenna looking from a certain direction (,) is the ratio of the average power P delivered to a matched load to the L magnitude of the average power density P of the avi incident electromagnetic wave at the position of the antenna multiplied by the normalized power pattern P (,) of that antenna. av P L A  ,  e P P,  avi av Hon Tat Hui Antenna Fundamentals 54NUS/ECE EE5308 The effective area is related to the directivity as (see Supplementary Notes): 2  AD,,     e 4 A maximum effective area A can be defined when the em antenna is receiving in its maximumdirectivity direction. That is, 2  AD em 0 4 Hon Tat Hui Antenna Fundamentals 55NUS/ECE EE5308 2.14 Open Circuit Voltage A receiving antenna can be modelled as an equivalent circuit as follows: I a L Incident wave  V oc I L R L a Equivalent R in Z  L b circuit X L X in b Receiving antenna a is positive with Z = R + jX = load impedance L L L respect to b Z = R + jX = internal impedance in in in Hon Tat Hui Antenna Fundamentals 56NUS/ECE EE5308 The opencircuit voltage V is defined as the voltage oc 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 positive V , I and E must oc i 1 be in opposite senses. Vd  IE  oc i  I m  where I current distribution on the atnenna when the antenna is excited at the terminal I current at the terminal m incident electricfield E i  length of the wire antenna Hon Tat Hui Antenna Fundamentals 57NUS/ECE EE5308 Proof of the OpenCircuit Voltage Expression Reciprocity Theorem  I dV 1 2 I m V  dI 1 2 Case 2 Case 1 I dI 12  VdV 12 Hon Tat Hui Antenna Fundamentals 58NUS/ECE EE5308 Putting VI Z , dV Ed 12 mA i we have, IEd I 1 i 1 dI dV 22 VIZ 1 mA 1 II  Ed   21 i  IZ mA  In vector form, 1 I  IE  d 21 i  IZ mA  Hon Tat Hui Antenna Fundamentals 59NUS/ECE EE5308 Putting I equal to I and note that I is the shortcircuit 1 2 current at the terminal of the antenna. By Thevenin’s theorem, the opencircuit voltage V at the antenna oc terminal can then be express as: 1 VIZ  IEd oc 2 A i  I m  (For a more detailed explanation on the reciprocity theorem, see Chapter 11, ref. 4.) Hon Tat Hui Antenna Fundamentals 60NUS/ECE EE5308 Example 8 A plane wave of frequency 1 GHz, having a peak value of the electric field intensity of E = 5μV/m, is incident 0 normally (electric field lines parallel to the dipole, i.e., zE ˆ E = ) on a halfwave thin dipole antenna. Determine the 0 i opencircuit voltage V induced on this antenna. oc I L a E i I I L V oc a Z Z L  L b Z A b Hon Tat Hui Antenna Fundamentals 61NUS/ECE EE5308 Solutions (1) Current distribution of a halfwave dipole antenna: Iz Iz ( ') (A), where ˆ Ik sin hz' forz' 0   m Iz ( ') I sink h z'   m Ik sin hz' forz' 0   m 2 kh ,   4 (2) Incident field: 6 Ez E z510 V/m 垐  i 0 Hon Tat Hui Antenna Fundamentals 62NUS/ECE EE5308 (3) Opencircuit voltage: /4 6 11 510 Vd  IE    IE d  Iz ''dz  oc i i   II I mm m /4  66 51 0 I 5 10 m  I  m 8 C 31 0 where   0.3 m 9 f 10 Hon Tat Hui Antenna Fundamentals 63NUS/ECE EE5308 References: 1. David K. Cheng, Field and Wave Electromagnetic, Addison Wesley Pub. Co., New York, 1989. 2. John D. Kraus, Antennas, McGrawHill, 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, PrenticeHall, ley, New York, 1998. 5. Fawwaz T. Ulaby, Applied Electromagnetics, PrenticeHall Inc., Englewood Cliffs, N. J., 1968. 6. Joseph A. Edminister, Schaum’s Outline of Theory and Problems of Electromagnetics, McGrawHill, Singapore, 1993. 7. Yungkuo Lim (Editor), Problems and solutions on electromagnetism, World Scientific, Singapore, 1993. Hon Tat Hui Antenna Fundamentals 64
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