Communication Networks 1 Principles of Mobile Communications University Duisburg-Essen WS 2003/2004 Page 1 N e v e r s t o p t h i n k i n g.
Wave Propagation Single- and Multipath Propagation Overview: Reminder: Basics of Vector Algebra Quantities of Electrodynamics Maxwell s Equations Electromagnetic Wave Equations Properties of EMW Basic Wave Propagation Mechanism - Reflection Page 2
Page 3
Principles of Mobile Communication Remember: Basics of Vector Algebra Maxwell * developed his original equations (1865) with the assumption that the electromagnetic fields need a invisible fluid (called aether) for their existence. 40 years later A. Einstein (1902) shows that the electromagnetic field exists without any fluid, also in vacuum (->no aether). (It needs again 50 years to distribute this to the experts...) Page 4 Therefore the vector algebra, developed and well known for fluids, was and is used to describe electromagnetic fields. *
Principles of Mobile Communication Definition of Fields: a) scalar field Basics of Vector Algebra a number can be assigned to all points in a room in a functional form. E.g. Temperature distribution, Pressure or Potential distribution a) vector field a vector can be assigned to a all point in a room in a functional form. E.g. force, velocity, electrical or magnetical field Page 5
Principles of Mobile Communication The gradient (grad): Basics of Vector Algebra Assume a scalar pressure field and a infinitely small volume (sphere) in the field. Then the gradient gives the direction and magnitude of the force trying to move the sphere into the direction with the lowest pressure. It gives the direction of the strongest increase or decrease of the scalar field. f = -grad p For electrical fields it gives the electrical field strength caused by a scalar potential field E = -grad Ψ p(x,y) y Page 6 (The gradient is transformation from scalar field to vector field.) x
Principles of Mobile Communication The divergence (div): Basics of Vector Algebra Assuming V is the velocity vector field of an incompressible streaming liquid. Then div V gives the sum of all liquid flowing in and out of a (infinitesimal) small volume at point V. div V = 0, there is no source and no sink in this field div V > 0 -> we have a source (source field) div V < 0 -> we have a sink (sink field) With other words: div V gives the source density of the field. Page 7 (The divergence is a transformation from a vector field to a scalar field.)
Principles of Mobile Communication Basics of Vector Algebra y Example: sink field div V < 0 Page 8 x Source: Extra lecture Physic B1, Prof. Dr. Metin Tolan, University Dortmund
Principles of Mobile Communication y Basics of Vector Algebra Example: no sink, no source: div V = 0 Page 9 x Source: Extra lecture Physic B1, Prof. Dr. Metin Tolan, University Dortmund
Principles of Mobile Communication The rotation (rot): Basics of Vector Algebra Assuming V is the velocity vector field of a incompressible streaming liquid. Then rot V gives the field describing the amount of liquid flowing tangential on the surface of an infinitesimal small volume at point V. rot V is again a vector field. (The rot operator transforms a vector field into another vector field.) Page 10
Principles of Mobile Communication Basics of Vector Algebra Example for rotation: Page 11 Source: Extra lecture Physic B1, Prof. Dr. Metin Tolan, University Dortmund
Principles of Mobile Communication Basics of Vector Algebra Using the nabla operator to show the parallelism between the vector operators. The nabla differential operator: (cartesian coordinates) Result: vector field Result: scalar field Result: vector field Page 12 nabla and grad is the same operator
Page 13 (cross product)
Basics of Vector Algebra Some more properties of vector algebra: (a rotated vector field where rot V is not zero is source free) Page 14 (a gradient field (potential field) is rotation free) (with vector fields V and W and scalar field f)
Wave Propagation Maxwell s Equation Electrodynamic Quantities for Field and Material:! E: electrical field strength [V/m]! D: electric displacement or electric flux density [As/m 2 ]! H: magnetic field strength [A/m] vector fields! B: magnetic induction or magnetic flux density [Vs/m 2 ]! J: electric current density [A/m 2 ]! ρ: electric charge density [As/m 3 ]! ε: electric permeability [As/Vm]! µ: magnetic permeability [Vs/Am] material parameter Page 15! κ: conductivity [A/Vm]
Wave Propagation Media (Material) Properties linear media: ε, µ, κ are independent from the field isotropic media: ε, µ, κ are independent from field direction homogeneous media: ε, µ, κ are independent from position in field dispersion free media: ε, µ, κ are independent from frequency loss-free media: κ = 0, ε r and µ r are real Page 16
Wave Propagation James Clerk Maxwell (1831-1879) Maxwell s equations in differential form André Marie Ampère (1775-1836) Michael Faraday (1791-1867) Page 17
Wave Propagation Media (Material) Properties The Maxwell equation summarizes the experimental laws of E&M. The equations were considered as the triumph of classical physics From these equations it was possible to show that they imply that light and other radiation's are propagating electromagnetic fields Page 18 The non-symmetry in Maxwell s equations may have disturbed many scientists -> A lot of extensions has been published in the last 100 years without success until now. -> Hertz Ansatz (total derivatives) -> Dirac Ansatz (magnetic monopoles) -> Harmuth Ansatz (no source field anymore) -> Munera-Guzman Ansatz (J, ρ are emf) -> Notation in Minkowski-Space (4-dimensional notation) -> simple Complex Notation -> Eight-dimensional complex notation The non-symmetrys in the Maxwell s equation still are correct.
Page 19 Field exists without current in media (J)
Wave Propagation Derivation of the wave equations - for charge, current and loss free media maxwell equation with rot operator vector algebra Gauss law time derivation of maxwell equation homogenous wave equation for H maxwell equation with rot operator Page 20 Gauss law homogenous wave equation for E
Wave Propagation Derivation of the wave equations For many practical important problems it is sufficient to regard harmonic processes only. For that we introduce the time independent phasors: Page 21
Wave Propagation Derivation of the wave equations Inserting the harmonic equations into the maxwell s equation yields to the Maxwell equation for phasors: Inserting the harmonic equations into the harmonic wave equations (here with J not zero) leads to the time independent wave equation for the phasors: Page 22
Wave Propagation Derivation of the wave equations General approach for planar waves traveling in z-direction Obviously the phasors are functions of z only, δ/δx=0 and δ/δy=0. Therefore Maxwell s equation are simplified to (J = 0): (no E z, H z ) Page 23,
Wave Propagation Derivation of the wave equations The homogenous wave equation simplifies to: Page 24 Dielectrical resistance: Free space resistance: (from Maxwell equations previoius slide)
Page 25
Wave Propagation Derivation of the wave equations Time shoot of a linear polarized electromagnetic wave: Page 26
Page 27
Page 28
Wave Propagation Basic Propagation Mechanisms The Three Basic Propagation Mechanisms Reflection wave impinges an object much larger than the wavelength Diffraction - radio path between receiver and transmitter is obstructed - surface of obstruction has sharp irregularities (edge) - depends on geometry, amplitude, phase, polarization Scattering - wave travels through medium with dimensions small compared to the wavelength - number of obstacles per unit volume is large Page 29
Page 30
plane (α e >brewster angle, E changes sign) Page 31
Page 32
Plane of incidence is the plane containing the incident, reflected and transmitted ray. (E-field in plane of incidence) (E-field normal to plane of incidence) (E-field in plane of incidence) (E-field normal to plane of incidence) Page 33
No reflected electrical field in the plane of incidence for vertical (i.e. parallel) Brewster-Angle polarization ( complete power goes into media 2) Page 34
Page 35 (α e is real)
α e =0 α e =90 Brewster angle Page 36
Page 37