12. 540 Principles of the Global Positioning System Lecture 16 Prof. Thomas Herring 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 1 12.540 Principles of the Global Positioning System Lecture 16 Prof. Thomas Herring
Propagation: ionospheric delay Summary Quick review/introduction to propagating waves Effects of low density plasma Additional effects Treatment of ionospheric delay in GPS processing Examples of some results 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 2 Propagation: Ionospheric delay • Summary – Quick review/introduction to propagating waves – Effects of low density plasma – Additional effects – Treatment of ionospheric delay in GPS processing – Examples of some results
Microwave signal propagation Maxwell's equations describe the propagation of electromagnetic waves( e.g. Jackson Classical Electrodynamics, Wiley, pp. 848 1975) 4丌r1aD V·D=4npV×H=-J+ at I aB V●B=0 V×E+ 0 c a 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 3 Microwave signal propagation • Maxwell’s Equations describe the propagation of electromagnetic waves (e.g. Jackson, Classical Electrodynamics, Wiley, pp. 848, 1975) ∇ • D = 4πρ ∇ × H = 4 π c J + 1 c ∂D ∂t ∇ • B = 0 ∇ × E + 1 c ∂B ∂t = 0
Maxwells equations In Maxwells equations E= Electric field; p=charge density; J=current density D= Electric displacement D=E+4TP where P is electric polarization from dipole moments of molecules Assuming induced polarization is parallel to E then we obtain D=EE. where s is the dielectric constant of the medium B=magnetic flux density(magnetic induction H=magnetic field; B=uH; u is the magnetic permeability 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 4 Maxwell’s equations • In Maxwell’s equations: – E = Electric field; ρ=charge density; J=current density – D = Electric displacement D = E+4 π P where P is electric polarization from dipole moments of molecules. – Assuming induced polarization is parallel to E then we obtain D = ε E, where ε is the dielectric constant of the medium – B=magnetic flux density (magnetic induction) – H=magnetic field; B = µ H; µ is the magnetic permeability
Maxwells equations General solution to equations is difficult because a propagating field induces currents in conducting materials which effect the propagating field Simplest solutions are for non-conducting media with constant permeability and susceptibility and absence of sources 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 5 Maxwell’s equations • General solution to equations is difficult because a propagating field induces currents in conducting materials which effect the propagating field. • Simplest solutions are for non-conducting media with constant permeability and susceptibility and absence of sources
Maxwell's equations in infinite medium With the before mentioned assumptions Maxwell's equations become laB V●E=0V×E+ a VoB=0 VXB-lECE c a Each cartesian component of E and B satisfy the wave equation 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 6 Maxwell’s equations in infinite medium • With the before mentioned assumptions Maxwell’s equations become: • Each cartesian component of E and B satisfy the wave equation ∇ • E = 0 ∇ × E + 1 c ∂B ∂t = 0 ∇ • B = 0 ∇ × B − µε c ∂E ∂t = 0
Wave equation Denoting one component by u we have C V--22=0 √AE The solution to the wave equation is ik.x-ior O u=e u8- wave vector E=F。kx-0B= vl XE 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 7 Wave equation • Denoting one component by u we have: • The solution to the wave equation is: ∇ 2 u − 1 v 2 ∂ 2 u ∂t 2 = 0 v = c µε u = eik.x −iωt k = ω v = µε ω c wave vector E = E 0 eik.x −iωt B = µε k × E k
Simplified propagation in ionosphere For low density plasma, we have free electrons that do not interact with each other The equation of motion of one electron in the presence of a harmonic electric field is given by m X+x+00x=-eE(x, t) Where m and e are mass and charge of electron and y is a damping force Magnetic forces are neglected 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 8 Simplified propagation in ionosphere • For low density plasma, we have free electrons that do not interact with each other. • The equation of motion of one electron in the presence of a harmonic electric field is given by: • Where m and e are mass and charge of electron and γ is a damping force. Magnetic forces are neglected. m Ý x Ý + γx Ý + ω0 2 [ x ] = − e E ( x,t )
Simplified model of ionosphere The dipole moment contributed by one electron Is p=-ex If the electrons can be considered free(Oo=0) then the dielectric constant becomes(with fo as fraction of free electrons 6()=E+i 4 rIFle mD(16-i0) 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 9 Simplified model of ionosphere • The dipole moment contributed by one electron is p=-e x • If the electrons can be considered free ( ω 0=0) then the dielectric constant becomes (with f 0 as fraction of free electrons): ε( ω) = ε 0 + i 4 πNf 0 e 2 m ω(γ 0 − i ω)
High frequency limit(GPS case When the EM wave has a high frequency, the dielectric constant can be written as for nz electrons per unit volume 4 INZe e()=1 plasma frequency For the ionosphere, Nz=104-106 electrons/cms and o p is 6-60 of mhz The wave-number is k=.|2a.|C O 04/08/02 12.540Lec16
04/08/02 12.540 Lec 16 10 High frequency limit (GPS case) • When the EM wave has a high frequency, the dielectric constant can be written as for NZ electrons per unit volume: • For the ionosphere, NZ=10 4-10 6 electrons/cm 3 and ω p is 6-60 of MHz • The wave-number is e ( ω) = 1 − ω p 2 ω2 ω p 2 = 4 πNZe 2 m ⇒ plasma frequency k = ω2 − ω p 2 / c