III. Spherical Waves and Radiation

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1 III. Spherical Waves and Radiation Antennas radiate spherical waves into free space Receiving antennas, reciprocity, path gain and path loss Noise as a limit to reception Ray model for antennas above a plane earth and in a street canyon Cylindrical waves Polytechnic University 00 by H. L. Bertoni 1

2 Radio Channel Encompasses Cables, Antennas and Environment Between Transmitting Antenna Receiving Antenna Radio Channel Tx Rx Information Cable Cable Information Transmitter impresses information onto the voltage of a high power RF carrier for transmission through the air - called modulation Receiver extracts the information from the voltage of a low power received signal - called demodulation Polytechnic University 00 by H. L. Bertoni

3 Examples of Different Cellular Antennas Half wave dipole λ / Full wave monopole above ground plane λ / λ /4 λ /4 Dipole in corner reflector Polytechnic University 00 by H. L. Bertoni 3

4 PCS Antennas Polytechnic University 00 by H. L. Bertoni 4

5 Base Station Antennas Polytechnic University 00 by H. L. Bertoni 5

6 Antennas Radiate Electromagnetic Waves EM waves have: Electric field E (V/m) Magnetic field H (A/m) E and H Transmitting Antenna Cable Perpendicular to each other and to direction of propagation - Polarization Amplitude depends on direction of propagation - Radiation Pattern H E Polytechnic University 00 by H. L. Bertoni 6

7 I x Spherical Waves Radiated by Antennas z φ θ I : terminal Current Z : constant with units of ohms η : 10π r E Radial Power Flux P = 1 Re E H 1 ZI { }= a r η r Antenna pattern = f θ,φ H a r y ( ) For large r, localized current sources radiate fields in the form of Spherical Waves E = a E ZI e jkr Polytechnic University 00 by H. L. Bertoni 7 r H = 1 η a r E a r = a E =1 f ( θ,φ) (watts/m ) f ( θ,φ)

8 Power Radiation Pattern P(θ) θ Power density radiated by antenna P(θ) = ExH* watts/m Poynting vector in the radial direction Polytechnic University 00 by H. L. Bertoni 8

9 Omnidirectional Antennas Polytechnic University 00 by H. L. Bertoni 9

10 Parabolic Reflector Antenna Polytechnic University 00 by H. L. Bertoni 10

11 Horn Antennas Polytechnic University 00 by H. L. Bertoni 11

12 Log Periodic Dipole Array Polytechnic University 00 by H. L. Bertoni 1

13 Dual Polarization Patch Antenna Polytechnic University 00 by H. L. Bertoni 13

14 P T = sphere P T = 1 η ZI Total Radiated Power P a r da, where da = r sinθ dθdφ π 0 π 0 f ( θ,φ) sinθ dθdφ P T is independent of r, as required by conservation of power. da r a r P ( ) Normalization for f θ,φ is: π 0 π 0 ( ) sinθdθdφ f θ,φ Then : P T = 4π ( ) = 4π η ZI f θ,φ and P = a r P T. 4πr 4πr = area of sphere Polytechnic University 00 by H. L. Bertoni 14

15 Antenna Gain and Radiation Resistance for No Resistive Loss Directive gain = g(θ,φ)= f (θ,φ) and Antenna gain = G = Max. value of g(θ,φ) If isotropic antennas could exist, then f (θ,φ) = 1, G = 1 Radiation Resistance R r = effective resistance seen at antenna terminals 1 R Z r I R r = P 4π = Z η = Rrη 4π T = 4π ZI η Polytechnic University 00 by H. L. Bertoni 15

16 Antenna Directivity, Gain, Efficiency Directivity = Gain = Efficiency = Maximum Pointing Vector Average Pointing Vector = P m (r) P ter min al 4πr ( ) P T P ter min al = P m (r) P T 4πr ( ) = P m (r) P av (r) includes the effect of antenna resistance Gain Directivity Polytechnic University 00 by H. L. Bertoni 16

17 Short (Hertzian) Dipole Antenna The radiated field can be L<<λ I z θ r E H z I (z) written in the desired form if E = a E ZI e jkr r sinθ f θ ()= 3 sinθ Starting with Maxwell's equations, it is found that E = a θ jη LI e jkr λ r sinθ Z = jη L 3 λ G = f ( 90 ) = 3/ R r = η π 3 L λ Polytechnic University 00 by H. L. Bertoni 17

18 Half Wave Dipole Antenna The radiated field can be written: z E λ /4 θ H I r I (z) λ /4 E = a E ZI where f θ ()= e jkr r f () θ cos π cosθ 0.781sinθ Z = j π η G = f ( 90 ) = logG =.db R r = 4π η j π η = 73 Ω Polytechnic University 00 by H. L. Bertoni 18

19 Summary of Antenna Radiation Radiation in free space takes the form of spherical waves E, H and r form a right hand system Field amplitudes vary as 1/r to conserve power Power varies as 1/r, and varies with direction from the antenna Direction dependence gives the directivity and gain of the antenna Radiation resistance is the terminal representation of the radiated power Polytechnic University 00 by H. L. Bertoni 19

20 Receiving Antennas and Reciprocity + V 1 - I 1 r I + V - For a linear two-port V 1 =Z 11 I 1 + Z 1 I V =Z 1 I 1 + Z I If I = 0, V = Z 1 I 1 ~ 1/r For r large, Z 1 << Z 11, Z + V 1 Reciprocity Z 1 = Z 1 Equivalent Circuit I 1 I + Z 11 -Z 1 Z -Z 1 Z 1 V Polytechnic University 00 by H. L. Bertoni 0

21 Circuit Relation for Radiation into Free Space Z 11 -Z 1 Z -Z I 1 V 1 Z 1 V (open circuit) - - V 1 = Z 11 I 1 V = V OC = Z 1 I 1 Transmitted power * P T = ( 1/)Re( V 1 I 1 )= ( 1/)Re( Z 11 I 1 )= (1 /)R r1 I 1 where R r1 = radiation resistance of antenna 1 Therefore : Z 11 = R r1 + jx 1 Similarily : Z = R r + jx where R r = radiation resistance of antenna Polytechnic University 00 by H. L. Bertoni 1

22 Received Power and Path Loss Ratio Z 11 -Z 1 + Z -Z I 1 V 1 Z 1 V - - I V Z * Matched Load R r - jx Z 1 Current I 1 divides between branches: I =-I 1 Z 1 + Z Z 1 +Z ( ) = -I 1 Z 1 R r Received Power for Matched Load P R = 1 I Rr = 1 I 1 Z 1 R r = I 1 Z 1 8R r Path Gain PG P R P T = I 1 Z1 8Rr I 1 Rr1 = Z 1 4R r1 R r Final expression for PG is the same if antenna radiates and antenna 1 receives. Polytechnic University 00 by H. L. Bertoni

23 Effective Area of Receiving Antenna Effective Area = A e g( θ,φ) P R = P a r A e = P T A 4π r e A e1 Z * P T Z * 11 P T A e PG = P R P T = g A e1 and by reciprocity PG = P R = g A 1 e 4 πr P T 4πr Therefore g A e1 = g 1 A e or A e1 g 1 = A e g = same for all antennas Polytechnic University 00 by H. L. Bertoni 3

24 Effective Area for a Hertzian Dipole z L<<λ I θ r I (z) E = a θ ZI e jkr r f () θ E θ g()= θ ( 3 ) ( sinθ) R r = η π 3 L λ + V OC = LE sinθ For matched termination + V oc - Z 11 Z11* or + V oc - R R RR + V oc / - P R = 1 V oc R R = V oc 8R R Polytechnic University 00 by H. L. Bertoni 4

25 Effective Area for a Hertzian Dipole - cont. For matched termination: P R = V OC LE sinθ = 8R R 8 η π = E 3 L η sin λ θ 4π = ( P a r)g(θ) λ 4π 3 λ In terms of the effective area P R = P a r A e. Comparing expressions, A e = g() θ λ 4π Polytechnic University 00 by H. L. Bertoni 5

26 Path Gain and Path Loss in Free Space For any antenna A e1 = A e = A = λ g 1 g g Hertz 4π Path gain in free space or A e = λ 4π g PG P R = g A 1 e = g A e1 λ = g P T 4πr 4πr 1 g 4πr For isotropic antennas g 1 = g =1 λ PG = 4πr Path Loss P T = 1 P R PG = 4πr λ 1 g 1 g Polytechnic University 00 by H. L. Bertoni 6

27 Path Gain in db for Antennas in Free Space PG db = PL db =10log g 1 g For isotropic antennas, g 1 = g =1 λ 4πr For frequency in GHz, λ = c f = 0.3 f GH PG db = 3.4 0log f GH 0log r PG db Slope=0 f GH = 1 r=1 r=10 r=100 r=1000 Polytechnic University 00 by H. L. Bertoni 7

28 Summary of Antennas as Receivers Directive properties of antennas is the same for reception and transmission Effective area for reception A e = gλ /4π For matched terminations, same power is received no matter which antenna is the transmitter Path gain PG = P R /P T < 1 Path loss PL = 1/PG > 1 Polytechnic University 00 by H. L. Bertoni 8

29 Noise Limit on Received Power Minimum power for reception set by noise and interference Noise power set by temperature T, Boltzman s constant k and bandwidth f of receiver: N = kt f For analog system, received power P R must be at least 10N For digital systems, the maximum capacity C (bits/s) in presence of white noise is given by the limit C = f log 1+ P R N Polytechnic University 00 by H. L. Bertoni 9

30 Sources of Thermal Noise Physical Temp of Line = T L Sky Temp ~5 o -150 o K T A Temp of Receiver T R Physical Temp of Antenna T AP Ground Temp ~300 o K Polytechnic University 00 by H. L. Bertoni 30

31 N = kt s f Thermal Noise Power N Boltsman s constant = k =1.38x10-3 watts/(hz o K) System temperature = T o S K Bandwidth = f Hz For T S = 300 o K and f = 30x10 3 Hz N = 1.4x10-16 watts (N) db = dbw = dbm Noise figure of receiver amplifier F ~ 5 db Effective noise = N + F For the example, N + F = dbm Polytechnic University 00 by H. L. Bertoni 31

32 WalkAbout Phones Frequency band Band width Thermal noise 4x10-18 mw /Hz Receiver noise figure SNR for reception Minimum received power Transmitted power Maximum allowed path loss Minimum path gain Antenna gain / Antenna height 450 MHz 1.5 khz 5x10-14 mw x10-1 mw 500 mw (P Tr ) db -(P Rec ) db P Rec /P Tr = Assume 0 db λ = m -133 dbm 5 db typical 10 db for FM -118 dbm 7 dbm 145 db 3.x m Polytechnic University 00 by H. L. Bertoni 3

33 Maximum Range WalkAbouts in Free Space PG = G 1 G or R < λ 4π λ 4πR = λ 4πR > = = m = 940 km or 563 miles Polytechnic University 00 by H. L. Bertoni 33

34 Summary of Noise Noise and interference set the limit on the minimum received power for signal detection Thermal noise can be generated in all parts of the communications system Miracle of radio is that signals ~ 10-1 mw can be detected Polytechnic University 00 by H. L. Bertoni 34

35 Ground and Buildings Influence Radio Propagation Reflection and transmission at ground, walls Diffraction at building corners and edges Diffraction Path Transmission Reflection Polytechnic University 00 by H. L. Bertoni 35

36 Two Ray Model for Antennas Over Flat Earth (Antennas are Assumed to be Isotropic) E 1 Antenna r 1 E h 1 r θ h Image θ λ P r = P t 4π α 1 exp( jkr r 1 )+Γθ () 1 exp( jkr 1 r ) Γ θ ()= cosθ a ε r sin θ cos θ + a ε r sin θ Polytechnic University 00 by H. L. Bertoni 36 R where θ = 90 α and a = 1 ε r for vertical (TM) polarization, or a = 1 horizontal (TE) polarization

37 Reflection Coefficients at Plane Earth Vertical (TM) and Horizontal (TE) Polarizations 1 Γ(θ ) Horiz. Pol. ε r =15-j0.1 Vert. Pol. ε r =15-j Incident Angle θ, degree Polytechnic University 00 by H. L. Bertoni 37

38 Path Gain vs. Antenna Separation (h 1 = 8.7 m and h = 1.8 m) Brewster s angle Path Gain (db) Vertical pol. Horizontal pol Γ = Distance (m) f = 900MHz Polytechnic University 00 by H. L. Bertoni 38

39 Sherman Island/Rural Polytechnic University 00 by H. L. Bertoni 39

40 Sherman Island Measurements vs. Theory Polytechnic University 00 by H. L. Bertoni 40

41 Flat Earth Path Loss Dependence for Large R If R >> h 1 and h then r 1, = R + ( h 1 m h ) R + 1 and Γ(θ ) -1 λ Received power P r = P t 4π ( ) = R + 1 R h 1 m h R h ( 1 + h )m h 1h R 1 exp( jkr r 1 )+Γθ () 1 exp( jkr 1 r ) is approximately P r = P t λ 4πR exp jk h 1h R exp jk h 1h R λ or P r = P t 4πR sin k h h 1 R λ = P t 4πR sin π h h 1 λr Polytechnic University 00 by H. L. Bertoni 41

42 Path Gain of Two Ray Model PG = λ 4πR sin π h 1h λr At the break point, R = 4h 1h λ λ PG = 4 4πR Past the break point λ PG 4πR πh 1h λr Past the break point, PG is : Independent of frequency Varies as 1 R 4 instead of 1 R. the path gain has a local maximum = h 1 h R 4 Polytechnic University 00 by H. L. Bertoni 4

43 Maximum Range for WalkAbouts on Flat Earth For h 1 = h =1.6 m, R B = 4h 1h λ = 4(1.6) =15.3 m For R > R B PG = (h 1 h ) R 4 > Solving the inequality for R R 4 ( ) < = or R < m = 5.3 km or 3. miles Polytechnic University 00 by H. L. Bertoni 43

44 Fresnel Zone Gives Region of Propagation r 1 r - r 1 = λ / r Fresnel zone is ellipsoid about ray connecting source and receiver and such that r -r 1 =λ/ Ray fields propagates within Fresnel zone Objects placed outside Fresnel zone generate new rays, but have only small effect on direct ray fields Objects placed inside Fresnel zone change field of direct ray Polytechnic University 00 by H. L. Bertoni 44

45 Fresnel Zone Interpretation of Break Point r 1 Fresnel zone (r - r 1 = λ /) r R B Fresnel zone definition : λ = r r 1 Horizontal antenna separation R B for Fresnel zone to touch the ground λ = r r 1 = R B + (h 1 + h ) R B + (h 1 h ) h 1h R B or R B 4h 1 h λ Polytechnic University 00 by H. L. Bertoni 45

46 Regression Fits to the -Ray Model on Either Side of the Break Point -50 Path Gain (db) f=1850mhz h 1 =8.7 h =1.6 Model: ray, ε r =15 n 1 =1.3 n = Distance (m) Polytechnic University 00 by H. L. Bertoni 46 R B

47 Six Ray Model to Account for Reflections From Buildings Along the Street z T α p R 0 R a Top view of street canyon showing relevant rays R b Each ray seen from above represents two rays when viewed from the side: 1. Ray propagating directly from Tx to Rx. Ray reflected from ground z R w Ray lengths : As seen from above R 0 = x + ( z T z R ) R a = x + ( w + z T + z R ) R b = x + ( w z T z R ) In 3D r n1, = R n + ( h 1 m h ) Polytechnic University 00 by H. L. Bertoni 47

48 Six Ray Model of the Street Canyon For x >> h 1,h polarization coupling at walls can be neglected. R Angle of incidence on ground θ n = arctan n h 1 + h For each ray pair (vertical polarization) V n = e jkr n1 r n1 +Γ H ( θ n ) e jkr n r n x Wall reflection angle ψ a,b = arctan w ± z T + z R Path Gain of six rays PG = λ 4π ( ) V 0 +Γ E ( ψ a )V a +Γ E ( ψ b )V b Polytechnic University 00 by H. L. Bertoni 48

49 Six Ray Model for Street Canyon f = 900 MHz, h 1 = 10 m, h = 1.8 m, w = 30 m, z T = z R = 8 m -40 Received Power (dbw) ray model 6 ray model Distance (m) Polytechnic University 00 by H. L. Bertoni 49

50 Received Signal on LOS Route f = 1937 MHz, h BS = 3. m, h m = 1.6 m Telesis Technology Laboratories, Experimental License Progress Report to the FCC, August, Polytechnic University 00 by H. L. Bertoni 50

51 Summary of Ray Models for Line-of-Sight (LOS) Conditions Ray models describes ground reflection for antennas above the earth Presence of earth changes the range dependence from 1/R to 1/R 4 Propagation in a street canyon causes fluctuations on top of the two ray model Fresnel zone identifies the region in space through which fields propagate Polytechnic University 00 by H. L. Bertoni 51

52 Cylindrical Waves Due to Line Source The concept of a cylindrical wave will be useful for discussing diffraction Phase is constant over the surface of a cylinder For ρ >> λ radiated fields are E = a E ZI e jkρ ρ f () θ y θ H ρ E x Line Source H = 1 η a ρ E z Field amplitudes vary as 1/ to conserve power. ρ Polytechnic University 00 by H. L. Bertoni 5

Problem 8.18 For some types of glass, the index of refraction varies with wavelength. A prism made of a material with

$Problem 8.18 For some types of glass, the index of refraction varies with wavelength. A prism made of a material with$ Problem 8.18 For some types of glass, the index of refraction varies with wavelength. A prism made of a material with n = 1.71 4 30 λ 0 (λ 0 in µm), where λ 0 is the wavelength in vacuum, was used to disperse