Satellite Communications Electromagnetic Wave Propagation • • • • • • • Overview Electromagnetic Waves Propagation Polarization Antennas Antenna radiation patterns Propagation Losses Goldstone antenna at twilight, NASA LECT 04 © 2012 Raymond P. Jefferis III 1 Reference Reference is specifically made to the following highly recommended source: Kraus, J. D. and Marhefka, R. J., Antennas For All Applications, Third Edition, McGraw-Hill, 2002 from which the antenna radiation equations used below were drawn. LECT 04 © 2012 Raymond P. Jefferis III 2 Overview • Satellite communication takes place through the propagation of focused and directed electromagnetic (EM) waves • Since both received and transmitted waves are simultaneously present at very different power levels, in a satellite, both frequency separation and EM field polarization are used to decouple the channels LECT 04 © 2012 Raymond P. Jefferis III 3 Maxwell’s Equations Maxwell’s equations in terms of free charge and current, WIKIPEDIA LECT 04 © 2012 Raymond P. Jefferis III 4 Wave Equation For scalar variable, u (E & M Fields) u 2 2 c(u) u 2 t 2 Solutions are sinusoids in time and space (waves) LECT 04 © 2012 Raymond P. Jefferis III 5 EM Wave Propagation Wikipedia • Electromagnetic (EM) waves propagate energy, contained in their electric and magnetic fields, through space with velocity v, which is the speed of light under the conditions of propagation. LECT 04 © 2012 Raymond P. Jefferis III 6 Transverse EM (Plane) Wave Properties • Velocity of propagation (near light speed) • Electric field is normal to the magnetic field • Both electric and magnetic fields are normal to direction of propagation (plane wave) • The relation of electric to magnetic fields is a constant for the medium (air, vacuum) • Waves are polarized, as determined by the direction of the electric field orientation LECT 04 © 2012 Raymond P. Jefferis III 7 Impedance • The electric field strength E and magnetic field intensity H in a propagating wave are related by, H 1 E where, LECT 04 = magnetic permeability [Henry/meter] 00-7 [Henry/meter] in vacuum = dielectric constant [Farads/meter] 0 = 1/36*10-8 [Farads/meter] in vacuum = impedance of the medium ( 0 =376.7 Ohms in free space) © 2012 Raymond P. Jefferis III 8 Impedance Change At Boundaries • At a boundary between two media of differing impedances (air and raindrops for instance), Z1 and Z2 [Ohms] – Part of the incident wave from Medium1 is reflected – Part of the incident wave is transmitted into Medium2 Z1 1 1 Z2 2 2 LECT 04 © 2012 Raymond P. Jefferis III 9 Wave Energy • The electric and magnetic energy densities in a plane wave are equal. [J/m2] • The total energy is the sum of these energies. [J/m2] LECT 04 1 2 wE E 2 1 wH H 2 2 wE w H wT wE wH wT 12 E 2 12 H 2 © 2012 Raymond P. Jefferis III 10 Wave Energy Density • The energy density of a plane wave is the Poynting energy, S [Watts/m2] SRMS 2 EH E E 1 2 1 1 2 1 2 SAv EH E E 2 2 2 LECT 04 © 2012 Raymond P. Jefferis III 11 Vertical Polarization Behavior • Radio frequency energy at frequency, f, propagates • The wave propagates away from the observer (into the paper), along the z-axis • Energy propagates with velocity, v, • As a function of distance, z, and time, t, the vertical electric field is described by, z E Ey Em cos 2 f t v LECT 04 © 2012 Raymond P. Jefferis III 12 Horizontal Polarization • Radio frequency energy at frequency, f , propagates • The wave propagates away from the observer, along the z-axis • Energy propagates with velocity, v, • As a function of distance, z, and time, t, the horizontal E-field is described by, z E Ex Em cos 2 f t v LECT 04 © 2012 Raymond P. Jefferis III 13 Manipulated Variable Example Run mCos example: • Vary the frequency and observe the results • Pick a position (say z = 0.5), and change the z-variable to see how the wave propagates past the selected location LECT 04 © 2012 Raymond P. Jefferis III Lect 00 - 14 Antennas • Electromagnetic circuits comparable in size to the wavelength of an alternating current • Have alternating electric and magnetic fields resulting in Electromagnetic (EM) radiation • Have a polarization specified by the electric field direction (horizontal or vertical) • Radiation pattern is affected by the shape of the current-carrying conductor(s) • The EM radiation propagates in space LECT 04 © 2012 Raymond P. Jefferis III Lect 00 - 15 Vertically Polarized Antenna • Total antenna length typically /2 • Electric field shown normal to the plane of the earth (vertical) • Oscillating electric fields produce accelerating and decelerating conduction electrons, with consequent radiation of EM-energy • A magnetic field surrounds the currentcarrying wire • The phases of the electric and magnetic fields differ by 90 degrees LECT 04 © 2012 Raymond P. Jefferis III 16 Horizontally Polarized Antenna • Total antenna length typically /2 where λ = c/f • Electric field shown parallel to the plane of the earth (horizontal) • Oscillating electric fields produce accelerating and decelerating conduction electrons, with consequent radiation of EM-energy • A magnetic field surrounds the currentcarrying wire • The phases of the electric and magnetic fields differ by 90 degrees LECT 04 © 2012 Raymond P. Jefferis III 17 Polarization Match Angles • A match angle, M, is defined as the angular polarization difference between a transmitting and a receiving antenna • Smaller match angles result in greater coupling between transmitting and receiving antennas • If the antennas are at opposite polarizations (vertical - horizontal) the received power will be zero, theoretically. LECT 04 © 2012 Raymond P. Jefferis III 18 Circular Polarization • Radio frequency energy at frequency, f, propagates as an EM wave, away from the observer, along the z-axis (into the paper) • The energy propagates with velocity, v • The electric and magnetic fields rotate in time (space) according to, LECT 04 Ex Em cos 2 f t z v Ey Em cos 2 f t z v 2 © 2012 Raymond P. Jefferis III 19 Circularly Polarized Antenna Circular Polarization, Wikipedia Note the spiral net electric field resolves into time-varying Ex and Ey components. Conductor (black); Ex => Green; Ey => Red LECT 04 © 2012 Raymond P. Jefferis III 20 The Isotropic (Ideal) Antenna • The gains of antennas can be stated relative to an isotropic ideal antenna as G [dBi], where G > 0. • This antenna is a (theoretical) point source of EM energy • It radiates uniformly in all directions • A sphere centered on this antenna would exhibit constant energy per unit area over its surface • The gain of an isotropic antenna is 0 dBi Lect 05 © 2012 Raymond P. Jefferis III Lect 00 - 21 Radiation Patterns of Antennas • Electric field intensity is a function of the radial distance and the angle from the antenna • A radiation pattern can be plotted to show field strength (shown as a radial distance) vs angle • The angle between half-power points (denoted as HPBW) is a measure of the focusing (Gain) of the antenna. [Note: Half-power = 3 dB] • Note: Antenna Gain is with respect to an ideal isotropic antenna (Gain = 1.0 or 0.0 dBi) LECT 04 © 2012 Raymond P. Jefferis III 22 Antenna Gain Calculation • G = PA/PI where, PI is the power per unit area radiated by an isotropic antenna, and PA is the antenna power per unit area radiated by a non-isotropic antenna, G is the amount by which the isotropic power would be multiplied to give the same power per unit area as the gain antenna exhibits in the chosen direction LECT 04 © 2012 Raymond P. Jefferis III 23 Antenna Gain Calculation • • • • Pr = radiated power per unit area W = total applied power Rr = antenna radiation resistance Im = maximum value of antenna current 4 r 2 Pr G W 2 Im W Rr 2 LECT 04 © 2012 Raymond P. Jefferis III 24 Antenna Gain and Aperture Calculations G 4 Ae Ae A LECT 04 2 G = antenna gain Ae = effective aperture area = carrier wavelength η = aperture efficiency A = aperture area (r2) © 2012 Raymond P. Jefferis III 25 Half-Wave Dipole Power cos cos Im 2 E 60 r sin 15I m 2 Pr r2 LECT 04 cos cos 2 sin θ is the angle normal to the antenna 2 © 2012 Raymond P. Jefferis III 26 Dipole Radiation Patterns • Two dipole lengths shown: L = /2 (half wave dipole) HPBW = 78˚ Gain = 2.15 dBi L = (full wave dipole) HPBW = 47˚ Gain = 3.8 dBi • The longer antenna focuses the energy into a more narrow beam and thus has higher Gain. Electric field intensity, half-wave dipole LECT 04 © 2012 Raymond P. Jefferis III 27 Half-Wave Dipole Radiation The radiated field and power of a half-wave dipole antenna are expressed by: cos cos 2 E sin P : E2 Radiated power pattern, half-wave dipole LECT 04 © 2012 Raymond P. Jefferis III 28 Half-Wave Dipole Radiation Pattern zro = 0.000001; e0 = 1.0; e1 = Cos[p/2*Cos[theta]]/Sin[theta]; e2 = e1^2; PolarPlot[{e2}, {theta, zro, Pi}, PlotStyle -> {Directive[Thick, Black]}, PlotRange -> Automatic] LECT 04 © 2012 Raymond P. Jefferis III 29 Half-Power Beam Width • The Half-Power Beam Width (HPBW) is defined as the included angle between the half-power points on the radiation pattern. The power is down by 3 dB at these points. • For a half-wave dipole antenna this is calculated as shown on the Mathematica® notebook output that continues below. LECT 04 © 2012 Raymond P. Jefferis III 30 Half-Wave Dipole HPBW Calculation r1 = FindRoot[e1^2 - 0.5 == 0.0, {theta, 60.0 Degree}]; Print[r1] w1 = theta /. r1 Print[w1/Degree] r2 = FindRoot[e1^2 - 0.5 == 0.0, {theta, 120.0 Degree}]; Print[r2] w2 = theta /. R2 Print[w2/Degree] Print[(w2 - w1)/Degree] LECT 04 © 2012 Raymond P. Jefferis III 31 HPBW for Half-Wave Dipole • From the foregoing notebook, the HalfPower Beam Width is found to be: HPBW = 78.0777 degrees • At the outer edges of the beam (HPBW), the power will be 70.7% of the maximum power value. LECT 04 © 2012 Raymond P. Jefferis III 32 Full-Wave Dipole Radiation The radiated field and power of a full-wave dipole antenna are expressed, as a function of angle, by: cos [ cos 1 E sin 2 P: E Power pattern, full-wave dipole LECT 04 © 2012 Raymond P. Jefferis III 33 Full-Wave Dipole Radiation Pattern zro = 0.000001; e0 = 1.0; en = 2.0; e1 = (Cos[p*Cos[theta]] + 1)/(Sin[theta]*en); e2 = e1^2; PolarPlot[{e2}, {theta, zro, p}, PlotStyle -> {Directive[Thick, Black]}] LECT 04 © 2012 Raymond P. Jefferis III 34 Half-Power Beam Width • The Half-Power Beam Width (HPBW) is defined as the included angle between halfpower points on the radiation pattern. The power is down by 3 dB at these points. • For a full-wave dipole antenna this is calculated as shown on the Mathematica® notebook output that continues below. LECT 04 © 2012 Raymond P. Jefferis III 35 Full-Wave HPBW Calculation r1 = FindRoot[e1^2 - 0.5 == 0.0, {theta, 60.0 Degree}]; Print[r1] w1 = theta /. r1 Print[w1/Degree] r2 = FindRoot[e1^2 - 0.5 == 0.0, {theta, 120.0 Degree}]; Print[r2] w2 = theta /. r2 Print[w2/Degree] Print[(w2 - w1)/Degree] LECT 04 © 2012 Raymond P. Jefferis III 36 HPBW for Full-Wave Dipole • From the foregoing notebook, the HalfPower Beam Width is found to be: HPBW = 47.8351 degrees • At the outer edges of the beam (HPBW), the power will be 70.7% of the full value. LECT 04 © 2012 Raymond P. Jefferis III 37 Circular Aperture Antenna • The electric field of a circular aperture antenna can be calculated from: 2 J1[( D / )sin ] E[ ] D sin where, D/ gives the aperture diameter in wavelengths and ϕ is the angle relative to the normal to the plane of the aperture. LECT 04 © 2012 Raymond P. Jefferis III 38 Radiated E-Field of Aperture Antenna 0.0 3 0.0 2 0.0 1 0.0 0 - 0.0 1 - 0.0 2 - 0.0 3 0.2 0.4 0.6 0.8 1.0 E-field for aperture with D/ = 10 The Mathematica® notebook follows, for D/ = 10: LECT 04 © 2012 Raymond P. Jefferis III 39 Radiation Pattern of Aperture Antenna Dlam = 10; e2 = (2.0/p*Dlam)*(BesselJ[1, p*Dlam*Sin[theta]])/Sin[theta]; PolarPlot[Abs[e2]/100, {theta, -p/6, p/6}, PlotStyle -> {Directive[Thick, Black]}] LECT 04 © 2012 Raymond P. Jefferis III 40 Radiated Power from an Aperture • The normalized radiated power can be found from E2[] as shown below: 0.04 0.02 0.00 - 0.02 - 0.04 0.2 0.4 0.6 0.8 1.0 Normalized radiated power for aperture with D/ = 10 LECT 04 © 2012 Raymond P. Jefferis III 41 Radiated Power Calculation Dlam = 10; e2 = (2.0/p*Dlam)*(BesselJ[1, p*Dlam*Sin[theta]])/Sin[theta]; PolarPlot[Abs[e2/100], {theta, -p/6, p/6}, PlotStyle -> {Directive[Thick, Black]}, PlotRange -> {{0, 1}, {-0.04, 0.04}}] LECT 04 © 2012 Raymond P. Jefferis III 42 Half Power Beam Width • The HPBW of an aperture having D/ = 10 is calculated to be: 5.89831 Degrees • The Mathematica® notebook for this calculation follows: LECT 04 © 2012 Raymond P. Jefferis III 43 Aperture HPBW Calculation p20 =((2.0/p*Dlam)* (BesselJ[1,p*Dlam*Sin[0.00001]])/Sin[0.00001])^2 p2 = ((2.0/p*Dlam)* (BesselJ[1,p*Dlam*Sin[theta]])/Sin[theta])^2/p20; r1 = FindRoot[p2 - 0.5 == 0.0, {theta,1 Degree}]; w1 = theta /. r1; Print[w1/Degree] r2 = FindRoot[p2 - 0.5 == 0.0,{theta,-1 Degree}]; w2 = theta /. r2 Print[w2/Degree] Print[Abs[(w2 - w1)]/Degree] LECT 04 © 2012 Raymond P. Jefferis III 44 Workshop 04 - Antenna HPBW • A circular aperture antenna has D/ = 20. Plot the radiation pattern of this antenna and calculate its Half Power Beam Width. • What can you say about the aiming requirements for such an antenna mounted on a satellite? LECT 04 © 2012 Raymond P. Jefferis III Lect 00 - 45 Transmission Losses Transmitted electromagnetic energy from a satellite is lost on its way to the receiving station due to a number of factors, including: – Antenna efficiency – Antenna aperture gain – Path loss LECT 04 – Rain/Cloud loss – Atmospheric loss – Diffraction loss © 2012 Raymond P. Jefferis III 46 Antenna Gain and Link Losses Pt = transmitted power Pr = received power At = transmit antenna aperture Ar = receive antenna aperture Lp = path loss La = atmospheric attenuation loss Ld = diffraction losses Antenna Gain (t or r): Gt/r = 4Ae t/r/ 2 Combined Antenna Gain (t + r): G = GtGr LECT 04 © 2012 Raymond P. Jefferis III 47 Antenna Gain Ae A (d / 2) G 4 2 Ae d G A LECT 04 2 2 Ae = effective antenna aperture G = 4Ae/ 2 (Antenna Gain) d = antenna diameter λ = wavelength = aperture efficiency © 2012 Raymond P. Jefferis III 48 Compensating for Link Losses • Increase antenna gain • Increase power input to antenna • Net effect: increase EIRP (Equivalent Isotropically Radiated Power) - Make sure tracking of beam is accurate (target on beam axis). LECT 04 © 2012 Raymond P. Jefferis III Lect 00 - 49 EIRP • Equivalent Isotropic Radiated Power • – the equivalent power input that would be needed for an isotropic antenna to radiate the same power over the angles of interest LECT 04 © 2012 Raymond P. Jefferis III Lect 00 - 50 Path Loss Calculation • Effective Aperture (transmit or receive): Ae = A • Effective Radiated Power: EIRP = PtGt = Pt tAt • Path Loss (for path length R): Lp = (4R/ 2 • Received Power: Pr = EIRP*Gr/Lp where, Gt = 4Aet/ 2 Gr = 4Aer/ 2 LECT 04 © 2012 Raymond P. Jefferis III 51 Decibel (dB) Scale Definition • PdB = 10 log10 Pt/Pr • Logarithmic scale changes division and multiplication into subtraction and addition • dBW refers to power with respect to 1 Watt. • Received power (Pratt & Bostian, Eq. 4.11): • Pr = EIRP + Gr - Lp [dBW] LECT 04 © 2012 Raymond P. Jefferis III 52 Received Power - dB Model • (Pratt & Bostian, Eq. 4.11) Pr = EIRP + Gr - Lp - La - Lt - Lr [dBW] – – – – – – LECT 04 EIRP => Effective radiated power Gr => Receiving antenna gain Lp => Path loss La => Atmospheric attenuation loss Lt => Transmitting antenna losses Lr => Receiving antenna losses © 2012 Raymond P. Jefferis III 53 End LECT 04 © 2012 Raymond P. Jefferis III 54