Ch3. Introduction To Antenna

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Ch3. Introduction to Antenna Chien-Wen Chiu(邱建文), Professor Department of Electronic Engineering National I-lan University 1, Sec. 1, Shern-Nong Rd., I-lan, Taiwan, R.O.C. e’mail address : [email protected] March 16, 2010 1 Content 1. Antenna introduction and parameters 2. Field for the Short Dipole Antenna 3. Near field and Far field 4. Radiation Patterns and Beamwidth 5. Antenna Gain, Directivity, and Efficiency 6. Antenna Bandwidth 7. Polarization 8. Input impedance 9. Max. Power Transfer and Effective Aperture 2 1. Antenna Introduction Example : Figure 13.1 (p. 634) by Pazar’s microwave enginnering Photograph of various millimeter wave antennas. Clockwise from top: a highgain 38 GHz reflector antenna with radome, a prime-focus parabolic antenna, a corrugated conical horn antenna, a 38 GHz planar microstrip array, a pyramidal horn antenna with a Gunn diode module, and a multibeam reflector antenna. 3 1. Antenna Introduction Basic operation of transmit and receive antennas. Figure 13.2 (p. 635) Basic operation of transmit and receive antennas. 4 5  c  f Frequency Band   c f  : wavelength f : frequency c: speed of light 6 Antenna - How it Works The antenna converts radio frequency electrical energy fed to it (via the transmission line) to an electromagnetic wave propagated into space. The physical size of the radiating element is proportional to the wavelength. The higher the frequency, the smaller the antenna size. Assuming that the operating frequency in both cases is the same, the antenna will perform identically in Transmit or Receive mode [email protected] 7 Antenna reciprocity theorem 發射至空中 來自空中 發射機天線 RF能量 RF能量 發射機 接收機 (a) (b) 天線的互易性定理(Antenna reciprocity theory): 任何天線， 若工作於相同之頻率，作用於發射端或接收端，當作發射天線或 接收天線都有相同之效率及特性，此種特性稱為天線之互易性。 8 Current distribution of transmission line and linear dipole 9 Sin(kl/2) ≒kl/2 as kl is very small Significant interference and cancelling will be noted 10 (900) (1800) 11 Current variation of as a function of time for half-wavelength dipole   E   j H Radiation Mechanism   H  j E D  0  E B  0  H Why ? How ? What ? 12 (900) (1800) 13 Radiation Mechanism for Dipole Antenna (2) (1) (1) (1) (2) (1) (1) (1) 傳播到空 中的電場 (a) (b) (c) (d) 14 (900) 15 Maxwell Equations      E   jH  M      H  jE  J  D    B  0 16 Types of antenna 1.Wire antennas (and loop antenna) 2. Aperture antennas 3. Microstrip antennas 4. Array antennas 5. Reflector antennas 6. Lens antennas 17 Antenna Parameters Frequency band Bandwidth S11(dB) or VSWR@50 Ohm Patterns Gain Efficiency Polarization Size 18 2. Field for the Short Dipole Antenna I o L 1 j  j r cos  [ 2  ] e 2 r r 3 I o L j 1 j  j r E  sin [  2  ] e 2 r r r 3 Io L j 1  j r H  sin [  2 ]e 4 r r 0 w    ,   c  0 0 Er  H r  H   0, E  0 How to obtain these equations? 19 The elemental electric dipole (詳細推導請參閱Cheng的電磁學第十章) R=r 20 Derive Formula Magnetic vector potential: [ I ] A dlaˆ z 4 r [ I ]  I o cos(t)  r ),   R=r 2   I o dl  j  r Az  aˆ z e , (phasor form) 4 r A  A( Ax , Ay , Az )  ( Ar , A , A ) For spherical coordinates: Ar  Az cos  , A   Az sin  , A  0 H  1   A I o dl j 1 sin  [  2 ]e  j  r 4 r r H r  H  0  H  21 E 1 j Derive Formula  H  I o dl 1 j  j r cos  [ 2  ] e 2 r  r3  I o dl j 1 j  j r E  sin  [  2  ] e 4 r r  r3  Er  E  0   w (Conduction field, similar to that of dipole, as β =0)  ,    c  I o dl 1  j r  Er  cos  [1  ] e 2 r 2 j r  I dl 1 1  j r E  j o  sin  [1   ] e 4 r j  r ( r )2  0 I o dl 1 H  j  sin  [1  ]e  j  r 4 r j r 22 (Conduction field, similar to that obtained by Biot-Sarvart theorm, as β0=0) 3. Near field and Far field for small dipole ( / 50  L   / 10) 1. Reactive near field region 2. Radiating near field region (Fresnel region) 3. Radiating far field region (Fraunhofer region) (have Er) 23 Near field and far field for infinitesimal dipole j 1 j  2  ) 3 r r r  1 1  2  r r  r3 E  (  r 1 r   2  I . r  (Near-field-region) 2  I o dl  j  j r Er  cosθ e 3 2 r (Conduction field, similar to that of dipole, as β =0)  I o dl  j  j r E  sinθ e 3 2 r 0 H  I o dl 1 sinθ 2 e  j  r 4 r (Conduction field, similar to that obtained by Biot-Sarvart theorm, as β0=0) 24 Near field and far field for infinitesimal dipole  Io L 1 j  j r cos  [ 2  ] e 2 r  r3  Io L j 1 j  j r E  sin  [  2  ] e 2 r r  r3 Io L j 1  j r H  sin  [  2 ]e 4 r r Er   II . r  (Far-field-region) 2  I o dl j   j r E  sinθ e 4 r (same phase, orthogonal direction) I o dl j   j r H  sinθ e 4 r Er  0, E / H    25 4. Antenna Radiation Pattern I. Radiation Pattern(2D or 3D)  A graphical representation of the intensity of the radiation vs. the angle from the perpendicular.  The graph is usually circular, the intensity indicated by the distance from the center based in the corresponding angle. [email protected] 26  I o dl j   j r E  sinθ e  E  sin  4 r H-plane : the plane containing the magnetic-field vector and the direction of maximum radiation E-plane : the plane containing the electric-field vector and the direction of maximum radiation The pattern of the H-field is same as that of E-field 27 28 The type of system you are installing will help determine the type of antenna used. Generally speaking, there are two „types‟ of antenna: 1. Directional - this type of antenna has a narrow beamwidth; with the power being more directional, greater distances are usually achieved but area coverage is sacrificed - Yagi, Panel, Sector and Parabolic antenna - Some time we will use this type of antenna in both Point to Point and Point to Multipoint communications 2. Omni-Directional - this type of antenna has a wide beamwidth and radiates 3600; with the power being more spread out, shorter distances are achieved but greater coverage attained 3. - Omni antenna [email protected] 29 Radiation Pattern (elevation) main lobe boresight side lobe Radiation Pattern [email protected] 30 Side lobes Upper Side Lobe Suppression (dB) [email protected] 31 Directive Radiation Pattern (Broadband)  Log periodic dipole array (LPDA) Directional Radiation Pattern Dipoles Transmission line - very wide BW, with constant SWR - typical gain 10 dBi main lobe • Reflector Yagi antenna Driven element (dipole) Directors back lobe side lobe - BW is smaller than LPDA - typical gain 12 – 14 dB [email protected] main lobe 32 Antenna Radiation pattern Directional Antenna Radiation Pattern Horizontal plane Vertical plane Horizontal-plane and Vertical plane : based on the earth [email protected] 33 Antenna Radiation pattern Omni-directional Antenna Radiation Pattern H-plane E-plane H-plane : the plane containing the magnetic-field vector and the direction of maximum radiation E-plane : the plane containing the electric-field vector and the direction of maximum radiation 34 (Horizontal Plane) (Horizontal Plane) Typical Radiation Pattern for a Yagi [email protected] 35 270 0 -3 -6 0 0 -15 -20 -15 -20 -30 -30 -10 dB 90 270 0 -3 -6 -10 dB 90 180 180 Typical Radiation Pattern for a Sector antenna [email protected] 36 Three-dimensional Pattern [email protected] 37 II. Power Patterns and Beamwidth Power Density:  * 1 Wav  Re[ E  H ] W/m 2 2 Power Pattern Normalized Power Pattern Beamwidth: -3dB : half-power beamwidth -- Half Power Beamwidth (HPF) -- Beam Efficiency (BE) 2 1 0 2 0 0 0 U ( , ) sindd   BE      U ( , ) sindd Beamwidth of half-wavelength dipole antenna is 78o 38 Beamwidth 10dB Beamwidth 3dB Beamwidth Peak - 10dB Peak - 3dB 60° (eg) Peak Peak - 3dB [email protected] 120° (eg) Peak Peak - 10dB 39 III. Power Density and Radiation Intensity A. Radiation Power Density Poynting vector: W  E  H * Watt / m 2 (Instantaneous Poynting vector) P  S W  dS   S ˆ (Instantaneous total power) W  nda Average Power Density: Wav ( x, y, z )  Wav ( x, y, z, t )   av Average Power: Prad  Pav   S Wav  dS  1 2  S 1 Re[ E  H * ] 2 Re[ E  H * ]  dS dS  r 2 sin  d d  S m2 (Time average) 1 Radiation Power density: Re[ E  H * ] 2 B. Radian & Solid angle solid angle: d   W dS  sin  d d r2 d     4 40 Power Density and Radiation Intensity C. Radiation Intensity U  r 2Wav W unit solid angle : far zone parameter 2 r2 r2  2 2  U ( ,  )  E (r , ,  )  E ( r ,  ,  )  E ( r ,  ,  )    2 2  2 1 2   U ( ,  )  U ( ,  )  , E (r )   r Prad    Ud    2 0   0 U sin  d d Example: Calculate the total radiated power sin  If an antenna has : Wav  aˆrWav  aˆr Ao 2 U=Aosin r 2  sin  Prad   Wav  dS    aˆr Ao 2  aˆr r 2 sin  d d   2 A0 (W ) S 0 0 r Prad    Ud    2 0   0 Ao sin  sin  d d   2 Ao (W ) D. Isotropic Source: Prad    U o d   4 U o  U o  Prad 4 41 Power Density and Radiation Intensity Example: A hypothetical isotropic antenna is radiating in free space. At a distance of 100 mm from the antenna , the total electric field E ( ) is measured to be 5 V/m. Find (a) the power density (Wrad) (b) the power radiated (Prad) 42 5. Antenna Directivity, Gain, and Efficiency Directivity : Maximum radiation intensity/Average radiation intensity U Umax : MAX radiation intensity D  rad U0 , U0 : radiation intensity of isotropic source U max 4 U max , P : radiation power from antenna rad =  Prad / 4 Prad , (U0 = Prad /4 ) (lossless isotropic source) Gain : 4 Maximum radiation intensity/Input Power G U max 4 U max  Pin / 4 Pin ,Prad : input power from the input port Efficiency : Gain/ Directivity ecd =Prad / Pin = G/D * A short dipole antenna : D = 1.5 = 1.76 dBi (10logD) A half-wavelength dipole antenna : D = 1.64= 2.15 dBi Relative Gain or Directivity : Gr(dBd) =G(dBi)-2.15 43 Directivity and gain definition D = U rad U0 U max 4 U max  Prad / 4 Prad 4 44 Gain Unless otherwise specified, the gain usually refers to the direction of maximum radiation. Gain of this direction [email protected] 45 Gain Unit Antenna gain is usually expressed in dBi or dBd dBi Gain relative to an isotropic antenna when the reference antenna is an isotropic antenna. dBd Gain relative to a half-wave dipole when the reference antenna is a half-wave dipole. [email protected] 46 dBd and dBi isotropic radiator 2.15dB eg: 0dBd = 2.15dBi half-wave dipole [email protected] 47 Dipoles Wavelength 1/4 Wavelength 1/2 Wavelength 1/4 Wavelength 1/2 Wavelength Dipole 1900MHz ：78.95mm 800MHz ：187.5mm [email protected] 48 Dipoles One dipole multiple dipoles Received Power：1mW Received Power ：4 mW GAIN= 10log(4mW/1mW) = 6dBd [email protected] 49 Sector antenna compared with the Dipoles Antenna (down look) Omnidirectional array Sector antenna Received Power ：1mW Received Power ：8mW 10log(8mW/1mW) = 9dBd [email protected] 50 Measurement at anechoic chamber GAUT = Gstandard + PAUT - Pstandard GAUT : Gain of AUT (dB) Gstandard : Gain of standard gain antenna (dB) PAUT : Measured power of AUT (dBm) Pstandard : Measured power of standard gain antenna (dBm) Standard gain antenna : 1. BBHA 9120 LFA 700MHz-6GHz 2. TDK 9120D Horn antenna (900MHz-18GHz) 3. EMCO 3115 Double-Ridged Horn(1-18GHz) 4. Spectrum Technology : DRH-0118(1-18GHz) (3D chamber) 51 Yagi - better suited for shorter links - lower dBi gain; usually between 7 and 15 dBi [email protected] 52 Parabolic - used in medium to long links - gains of 18 to 28 dBi - most common [email protected] 53 Sector antenna(扇型天線) -A sector antenna is a kind of directional antenna with a sectorshaped radiation pattern. In mobile communications, these antennas are typically installed in base station sites for point-to-multipoint connections - directional in nature, but can be adjusted anywhere from 450 to 1800 - typical gains vary from 10 to 19 dBi 0 -15 -20 -30 270 0 -3 -6 -10 dB 90 180 [email protected] 54 Omni - used at the some communications for wide coverage - typical gains of 3 to 10 dBi 55 Directivity A. Directivity : the radiation intensity in a given direction the radiation intensity averaged over all directions U Wav 4 U D   U o Wave Prad If the directivity is not specified, it implies the direction of maximum radiation intensity. Dmax  D  U max Uo  U max 4 U max  Uo Prad 4 U  4 U , D  ( Prad )  ( Prad ) ( Prad )  ( Prad ) Example: (linear dipole l<<) (Homework 3.1) if : Wav  aˆr Ao sin 2  / r 2 (W / m 2 ) 3 Please proof : Dmax  , D( ,  )  1.5sin 2  2 56 Gain & Antenna Efficiency Radiation intensity U ( ,  ) U ( ,  ) G  4  4  Total input power Pin ( Pin / 4 ) Prad  ecd Pin  G ( ,  )  ecd D ( ,  ), (ecd obtained by measurement G  G ( ,  ) max  ecd D( ,  ) max  ecd D G (dB)  G (dB)  G (dB) G (dB)  10 log10 (ecd D) 57 Gain & Antenna Efficiency   Antenna Efficiency er  (1   ) : Reflection (Mismatch) efficiency 2 ec : Conductivity efficiency ed : Dielectric efficiency ecd  ec ed Antenna radiation efficiency eo  er ec ed  ecd (1   ) : 2 Total efficiency 58 Gain Example Example: A lossless dipole antenna, with input impedance 73 Ohms, is to be connected to a transmission line whose characteristic impedance is 50 Ohms. Assume that the pattern of the antenna is given approximately by : U ( , )  Bo sin 3  Find the overall maximum gain of this antenna. 2  U max  Bo  Prad    U ( ,  ) sin  d d 0 0   Prad D  4 3 2  2 Bo  sin  d  Bo ( ), 4 0 4 U max 16   1.697 Prad 3 ecd  1  G  ecd D  1.697  2.297( dB ) 73  50 2 )  0.965 73  50  er  0.965  0.155 dB er  (1   )  1  ( 2 eo  ecd G e eo cd D  0.965  1.697  2.142 (dB) 59 Gain (Homework 3.4) A lossless resonant half-wavelength dipole antenna, with input impedance of 100 ohms, is to be connected to a transmission line whose characteristic impedance is 25 ohm. Assuming that the pattern of the antenna is given approximately by U  Bo sin3  find the overall maximum gain of this antenna. Answer : 2.997 (dB) [email protected] (Refer to p.61of Balanis‟s “antenna theory”) 60 Average Gain & Antenna Efficiency for portable device   PeakGain : Radiation intensity U ( ,  ) U ( ,  ) G  4  4  Total input power Pin ( Pin / 4 )   Mean gain : (Average Gain) calculted by averaging the measred gain at sufficient points on a (typical spherical) surface around the handset. If the antenna was lossless, the mean gain woud be 0dBi.  3D Average gain  10 log10  (3-dimensional average gain) 3D Average gain=0 dB   =100%   Two-dimensional average gain (Gavg ) for notebook computer: N Gavg  10 log10  G ( )  G ( ) i 1 h i N v i [from 0 to 3600 , Npoints] 61 6. Antenna Bandwidth B  f H  f L 0 -5 1. GSM: 880-960 MHz -10 2. DCS : 1710-1880 MHz -15 -20 3. PCS : 1850-1990 MHz 4. WLAN/BlueTooth : 2400-2484 MHz fH fL -25 -30 5. GPS : 1575.42 ±1.023 MHz 0.7 0.8 0.9 1.0 6. W-CDMA : 1.920 -2.170 GHz Return Loss : 7. CDMA : 869-894 MHz(Qualcomm) ( RL )  20 log  1.1 一般天線頻寬與反射係數的關係是取小S11=-10dB以下的頻率範圍當作頻 寬，當小於-10dB以下的頻寬時，其反射係數=1/3、駐波比VSWR=2:1， 表示此天線在此頻率範圍至少有90%以上的能量輻射出去。 Broadband antenna : eq. 10:1 Narrowband antenna : eq.10% fraction bandwdith= B *100% fc 62 VSWR＝ VSWR  Zin  Z 0 100  50   1/ 3 Zin  Z 0 100  50 Vmax Vmin  1  1  (1 ) forward: 10W 100 ohms 50 ohms reverse: 1W 9W Return Loss：-20log(1/3) ≒ 10 dB VSWR (Voltage Standing Wave Ratio) Vr2 / Z 0 Vr    2 Vin Vin / Z 0 1  VSWR＝ ＝2 1  Pr 1   1/ 3 Pin 10 Usual Request：VSWR2.0 Reflection Coefficient：=(VSWR-1)/(VSWR+1) Return Loss：RL=-20lg  63 7. Polarization An antenna polarization is relative to the E-field of antenna. – If the E-field is horizontal, than the antenna is Horizontally Polarized. – If the E-field is vertical, than the antenna is Vertically Polarized. No matter what polarity you choose, all antennas in the same RF network must be polarized identically regardless of the antenna type. [email protected] 64 7. Polarization Polarization: -- Polarization of an antenna: the polarization of the wave transmitted by an antenna. -- Polarization of radiated wave: -- Electric-field : Time varied. A fixed point in space. -- Linear: a function of time along a line. Vertical polarization Horizontal polarization -- Circular: CW(clockwise), CCW(counterclockwise). -- elliptical: -- Co-polarization: -- Cross polarization: E = E e(jwt+1) E = E e(jwt+2) E-field direction : Vertical Polarization Horizontal Polarization Circular Polarization 65 Polarization Vertical Horizontal Vertical Polarization: The electric field is vertical to the ground (In the maximum gain direction) Horizontal Polarization: The electric field is parallel to the ground (In the maximum gain direction) [email protected] 66 Polarization + 45degree slant - 45degree slant [email protected] 67 Polarization V/H (Vertical/Horizontal) Slant (+/- 45° ) [email protected] 68 Polarization may deliberately be used to: – Increase isolation from unwanted signal sources (Cross Polarization Discrimination (x-pol) typically 25 dB) – Reduce interference – Help define a specific coverage area Horizontal Vertical [email protected] 69 Polarization of plane wave 1. Linearly polarization : E ( z )  aˆ x E x ( z ) 2. Consider the superposition of linearly polarized wave: E ( z )  aˆ x E x ( z )  aˆ y E y ( z )  aˆ x E xo e  jkz  aˆ y E yo e  jkz  e  j / 2   E ( z , t )  Re [ aˆ x E x ( z )  aˆ y E y ( z )]e jwt  aˆ x E xo cos(t  kz )  aˆ y E yo cos(t  kz   Set z  0 E (0, t )  aˆ x E x (0, t )  aˆ y E y (0, t )  aˆ x E xo cos(t )  aˆ y E yo sin(t ) As wt increases from 0 through  /2, the tip of the vector will have an locus. E (0, t ) cos(t )  x E xo sin(t )  E y (0, t ) E yo  E (0, t )   1  cos 2 (t )  1   x   E xo  2 2  E y (0, t )   E x (0, t )   1    E E   xo yo   2.1 Circular polarization:Exo  E yo 2 2.2 Elliptical polarization:E xo  E yo 70 2 ) 71 Electrical Field Representation  E ( z , t )  aˆ x E x ( z , t )  aˆ y E y ( z , t ) E x ( z , t )  Re[ E x e j (t  kz ) ]  Re[ E xo e j (t  kz  x ) ]  E xo cos(t  kz   x ) E y ( z , t )  Re[ E y e j (t  kz ) ]  Re[ E yo e j (t  kz  y ) ]  E yo cos(t  kz   y ) For linear polarizati on :    y   x  n For circular polarizati on : E xo  E yo When    y   x   ( 2n  12 ) , n  0,1,2,  for CW    y   x  ( 2n  12 ) , n  0,1,2,  for CCW For elliptical polarizati on : E xo  E yo When or    y   x   ( 2n  12 ) , n  0,1,2,  for CW    y   x  ( 2n  12 ) , n  0,1,2,  for CCW    y   x   n  , n  0,1,2  ,  0, for CW 2  0, for CCW 72 Derive Formula E x  E xo cos(t  kz   x )  E xo cos(t   xo ) E y  E yo cos(t  kz   y )  E yo cos(t   yo ) Ex  cos t cos  xo  sin t sin  xo , (1) E xo Ey E yo  cos t cos  yo  sin t sin  yo , (2) (1) sin  yo  (2) sin  xo  sin  yo Ey Ex  sin  xo  cos t (cos  xo sin  yo  cos  yo sin  xo ) E xo E yo  cos t sin  , (1) cos  yo  (2) cos  xo  cos  yo Ey Ex  cos  xo  sin t (cos  xo sin  yo  cos  yo sin  xo ) E xo E yo  sin t sin  ,   1 sin 2  1 sin 2  [(sin  yo [(    yo   xo    yo   xo Ey 2 Ey 2 Ex E  sin  xo )  (cos  yo x  cos  xo ) ] 1 E xo E yo E xo E yo Ex E y Ey 2 Ex 2 )  2 cos  ( ) ] 1 E xo E xo E yo E yo 73 Derive Formula Y Y‟ x  x' cos   y ' sin  ; y  x' sin   y ' cos  ( A cos 2   B cos  sin   C sin 2  ) x'2  (2 A cos  sin   2C sin  cos   B (cos   sin  ) x' y ' 2 2 X‟  x  ( A sin 2   B cos  sin   C cos 2  ) y '2  k 1 B  C  A sin 2  B cos 2  0    tan 1 ( ) 2 AC 1 1 1 A 2 , B  2 cos  , C 2 E xo E xo E yo E yo   2 E xo E yo 1 tan 1 ( 2 cos  ) 2 2 E xo  E yo Major axis : 2 4 2 OA  { 12 [ E xo2  E yo  ( E xo4  E yo  2 E xo2 E yo cos 2 ) 2 ]}2 1 1 2 4 2 OB  { 12 [ E xo2  E yo  ( E xo4  E yo  2 E xo2 E yo cos 2 ) 2 ]}2 1 Axial Ratio : OA major axis  OB min or axis 1 ,1  AR   74 Polarization Loss Factor  Incoming wave (electric field)  Ei  ˆ t Ei  The receving antenna :  Er  ˆ r Er Ei  aˆ x Eo ( x, y )e  jkz , Er  (aˆ x  aˆ y ) E ( x, y )e  jkz ˆt  ?, ˆ r  ?, PLF  ? Polarization Loss Factor : PLF  ˆ t  ˆ r  cos 2 Example : Linear polarization: 2 solve: ˆ t  aˆ x , ˆ r  1 (aˆ x  aˆ y ) 2 2 Polarization efficiency :  PLF (dB )  10 log10 ( ˆ t  ˆ r )  3dB   2 le  Einc Pe   2  2  Receiving mode le Einc   Transmitting mode le : Vector effective length of the antenna.  Einc : Incident electric field. 75 8. Input impedance of an Antenna Z A  RA  jX A ,   RA  Rr  RL R = 2Pt/I2 : Input resistance 2 Prad Rr  2 I0 : Radiation resistance RL  Loss (equivalent) Resistance due to skin effect or dielectric loss (heat) 76 Radiation Resistance for Infinitesimal Dipole 1 2 1 2 For Hertzian dipole: W  [ E  H *]  [aˆr E H *  aˆ Er H * ] (Poynting vector)  I ol sin 2  1 Wr  [1  j ] 2 8  r (  r )3 2  I ol cos  sin  1 W  j [1  ]  16 2 r 3 ( r )2 2 P  S W  dS   2 0   0 (aˆrWr aˆ W )  aˆr r 2 sin  d d  I ol 1  [1  j ] 3  (  r )3 2  Prad  j 2 (Wm  We ) ~ 2We 1 Q  Prad ( kr) 3 2 2 Prad 2 l Rr  2  80 , Io 2 Example : (Infinitesimal dipole) l  0.01 , l  2  Rr  73 (l   ) 50 (  120 ) 1 2 Rr  80 2 ( )  0.079 Ohm 100 77 Radiation Resistance for a Small Dipole Rrad : Radiation resistance RL : Loss due to material and metal G  ecd D, ecd  Rrad Rrad  RL -- From the circuit point of view:1. Power loss --- R 2. Stored energy --- L,C For Small Dipole:( /50