Antennas and Propagation

Transcription

1 Antennas and Propagation Ranga Rodrigo University of Moratuwa October 20, 2008 Compiled based on Lectures of Prof. (Mrs.) Indra Dayawansa. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

2 Tx Guided EM wave Rx Transmission line (a) Guided or wire-line communication. Tx Unguided EM wave Rx Transmission line Transmission line (b) Radio or wireless communication. Figure 1: Unguided and guided EM wave propagation. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

3 Part I Preliminaries Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

4 Scalar Fields Fields Scalar and Vector Fields The height of the surface of a conical pyramid versus position on its base is an example of a scalar field involving two variables. h(x, y) = 6 2 x 2 y 2 The distance field of points in a rectangular room is a scalar field in three dimensions. The temperature field associated with points in a room, when it is heated or cooled, is a time-varying scalar field. T(x, y, z, t). Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

5 Vector Fields Fields Scalar and Vector Fields To describe a vector field, we attribute to each point in the field region a vector that represents the magnitude and direction of the physical quantity under consideration at that point. Since a vector at a given point can be expressed as the sum of its components along a set of unit vectors at that point, a mathematical description of the vector field involves simply the descriptions of three component scalar fields. F(x, y, z, t) = F x (x, y, z, t)i x + F y (x, y, z, t)i y + F z (x, y, z, t)i z. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

6 Fields Scalar and Vector Fields Sinusoidally Time-Varying Fields If we have a sinusoidally time-varying scalar field, we can visualize the field quantity sinusoidally with with time at each point in the field region with the amplitude and phase governed by the spatial dependance of the field quantity. For example, the field Ae αz cos(ωt βz), where A, α and β are positive constants, is characterized by sinusoidal time variations with amplitude decreasing exponentially with z and the phase at any given time decreasing linearly with z. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

7 Fields Scalar and Vector Fields x t A sinusoidally time-varying field at a particular point in space. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

8 Fields Scalar and Vector Fields Sinusoidally Time-Varying Vector Fields The behavior of each component of the field may be visualized in the manner we discussed. If we fix our attention on a particular point in the field region, we can visualize the sinusoidal variation with time of a particular component at that point by a vector changing it magnitude and direction. If the tip of the vector simply moves back and forth along a line which is parallel to, say, x-axis, the component vector is said to be linearly polarized in the x-direction. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

9 Fields Scalar and Vector Fields Polarization of Sinusoidally Time-Varying Fields When two-component sinusoidally time varying vectors at a point are added, the polarization of the resulting sinusoidally time-varying vector can be 1 linear, 2 circular, or 3 elliptical. That is the tip of the vector can describe a straight line, circle or an ellipse with time, depending on the relative amplitude and the phase angles of the component vectors. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

10 Fields Linear Polarization Scalar and Vector Fields If two components sinusoidally time-varying vectors have arbitrary amplitudes, but are in phase as, for example, F 1 = F 1 cos(ωt + φ)i x, and F 2 = F 2 cos(ωt + φ)i y. then the sum vector F = F 1 + F 2 is linearly polarized in a direction making an angle α = tan 1 F y F x = tan 1 F 2 F 1. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

11 Fields Scalar and Vector Fields t t t Linear, circular, and elliptical polarizations. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

12 Fields Scalar and Vector Fields Two sinusoidally time-varying components are of arbitrary amplitudes but in phase are equal in amplitudes, and differ in phase by π/2 do not satisfy above two conditions Polarization linear circular elliptic Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

13 Fields Scalar and Vector Fields Polarization and Reception We generally refer to the electric field of the wave. If the incoming signal is linearly polarized, then for maximum voltage to be induced in a linear receiving antenna, the antenna must be oriented parallel to the direction of polarization of the signal. If the incoming signal is circularly or elliptically polarized, a voltage is induced in the antenna except for one orientation which is along the line perpendicular to the plane of the circle or of the ellipse. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

14 Fields The Electric Field Electric Field If we consider two point charges Q 1 C and Q 2 C separated by R m in free space, then the forces F 1 and F 2 experienced by Q 1 and Q 2, respectively, are given by F 1 = Q 1Q 2 4πε 0 R 2i 21. (1) F 2 = Q 2Q 1 4πε 0 R 2i 12. (2) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

15 Point Charges Fields Electric Field i 12 Q 2 F 2 R Q 1 F 1 i 21 Figure 2: Two point charges, forces, and unit vectors. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

16 Fields Electric Field Electric Field Intensity In in a region of space a test charge q experiences a force F, then the region is said ot be characterized by an electric field of intensity E given by The electric field intensity is given by E = F q. (3) F E = lim q 0 q. (4) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

17 Maxwell s Equations in Integral Form Faraday s Law The electromotive force around a closed path C is equal to the negative of the time rate of change of the magnetic flux enclosed by that path; that is, E dl = d B ds. (5) C dt S Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

18 Maxwell s Equations in Integral Form Ampère s Circuital Law The magnetomotive force around a closed path C is equal to the sum of current enclosed by that path due to theactual flow of charges and the displacement current due to the time rate of change of the displacement flux enclosed by that path; that is, H dl = J ds + d D ds. (6) C S dt S Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

19 Maxwell s Equations in Integral Form Gauss Law for the Electic Field The displacement flux emanating from a closed surface S is equal to the charge enclosed by that surface; that is, D ds = ρdv. (7) S V Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

20 Maxwell s Equations in Integral Form Gauss Law for the Magnetic Field The magnetic flux emanating from a closed surface S is equal to zero; that is, B ds = 0. (8) S Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

21 Maxwell s Equations in Integral Form Auxiliary Equation: The Law of Conservation of Charges An auxiliary equation, the law of conservation of charges, is given by J ds = d ρdv. (9) S dt V The current due to flow of charges emanating from a closed surface is equal to the time rate of decrease of the charge enclosed by that surface. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

22 Maxwell s Equations in Integral Form Other Relations D = εe, (10) H = B µ, (11) where ε and µ are the permittivity and permeability, respectively, of the medium. In addition, if the current density J is due to the conduction, then J = J c = σe, (12) where σ is the conductivity of the medium. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

23 Maxwell s Equations in Integral Form Boundary Conditions Maxwell s equations can be applied to closed paths and closed surfaces encompassing the boundary between two media, and in the limits the areas enclosed by the closed paths and the volumes bounded by the closed surfaces go to zero. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

24 Maxwell s Equations in Integral Form Boundary Conditions The boundary conditions are given by i n (E 1 E 2 ) = 0, (13) i n (H 1 H 2 ) = J s, (14) i n (D 1 D 2 ) = ρ s, (15) i n (B 1 B 2 ) = 0. (16) where the subscripts 1 and 2 refer to media 1 and 2, respectively, and i n is unit vector normal to the boundary at the point under consideration and directed into the medium 1. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

25 Maxwell s Equations in Integral Form Boundary Conditions At a point on the boundary, the tangential components of E and the normal components of B are continuous, whereas the tangential components of H are discontinuous by the amount equal to J S at that point, and the normal components of D are discontinuous by the amount equal to ρ S at that point. i n (E 1 E 2 ) = 0 i n (H 1 H 2 ) = J s i n (D 1 D 2 ) = ρ s i n (B 1 B 2 ) = 0 Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

26 Maxwell s Equations in Integral Form Maxwell s Equations in Integral Form: Summary E dl = d B ds. C dt S H dl = J ds + d C S dt D ds = ρdv. S V B ds = 0. S S D ds. (17) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

27 Maxwell s Equations in Differential Form Introduction The quantities involved in formulation of Maxwell s equations in integral from are the scalar quantities, electromotive force, magnetomotive force, magnetic flux, displacement flux, charge and current. These are related to the field vectors and source densities through line, surface, and volume integrals. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

28 Maxwell s Equations in Differential Form Introduction Thus the integral forms of Maxwell s equations, while containing all the information pertinent to the interdependence of field and source quantities over a given region of space, do not permit us to study directly the interaction between the field vectors and their relationships with the source densities at individual points. Differential forms of Maxwell s equations, on the other hand, apply directly to the field vectors and source densities at a given point. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

29 Maxwell s Equations in Differential Form The Curl The basic definition of curl is A = lim S 0 C A dl S i n. (18) max The expansion of curl in Cartesian coordinates is i x i y i z A = x y z. (19) A x A y A z Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

30 Maxwell s Equations in Differential Form The Divergence The basic definition of divergence is A = lim A ds S v 0 v. (20) The expansion of divergence in Cartesian coordinates is A = A x x + A y y + A z z. (21) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

31 Maxwell s Equations in Differential Form Faraday s Law The curl of the electric field intensity is equal to the negative of the time derivative of the magnetic flux density; that is, E = B t. (22) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

32 Maxwell s Equations in Differential Form Ampère s Circuital Law The curl of the magnetic field intensity is equal to the sum of current density due to flow of charges and the displacement current density, which is the time derivative of the displacement flux density; that is, H = J + D t. (23) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

33 Maxwell s Equations in Differential Form Gauss Law for the Electric Field The divergence of the displacement flux density is equal to the charge density; that is, D = ρ. (24) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

34 Maxwell s Equations in Differential Form Gauss Law for the Magnetic Field The divergence of the magnetic flux density is equal to zero; that is, B = 0. (25) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

35 Maxwell s Equations in Differential Form Auxiliary Equation: The Continuity Equation This equation, the differential form of the law of conservation of charges states that the sum of divergence of the current density due to flow of charges and the time derivative of the charge density is equal to to zero. J = ρ t. (26) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

36 Maxwell s Equations in Differential Form Other Relations Hold D = εe, (27) H = B µ, (28) where ε and µ are the permittivity and permeability, respectively, of the medium. In addition, if the current density J is due to the conduction, then J = J c = σe, (29) where σ is the conductivity of the medium. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

37 Maxwell s Equations in Differential Form Maxwell s Equations in Differential Form: Summary E = B t. H = J + D t. D = ρ. B = 0. (30) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

38 Maxwell s Equations in Differential Form Maxwell s Equations in Free Space E = B t. H = D t. D = 0. B = 0. (31) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

39 Maxwell s Equations in Differential Form Example 3.1 Given E = E m sin(ωt βz)i y in free space, find D, B, and H. Sketch E and H at t = 0. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

40 Maxwell s Equations in Differential Form Solution: The Maxwell equation E = B/ t gives i x i y i z x y z = B 0 E m sin(ωt βz) 0 t. or B = βe m cos(ωt βz)i x. t Integrating B = βe m ω sin(ωt βz)i x, where the constant of integration, which is a static field, has been neglected. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

41 Maxwell s Equations in Differential Form Then, H = βe m ωµ 0 sin(ωt βz)i x, E and H are mutually perpendicular. At t = 0, sin(ωt βz) = sin(βz). With this information, we can sketch E and H at t = 0. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

42 Wave Equations With the assumptions of region of interest being free of charge (ρ = 0), and isotropic materials with D = εe, B = µh, and J = σe, and time dependance e jωt for both E and H, Maxwell s equations become E = jωµh. H = (σ + jωε)e. D = 0. B = 0. (32) Taking the curl of the first two equations and using the identity ( A) = ( A) 2 A, Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

43 Wave Equations 2 E = jωµ(σ + jωε)e γ 2 E 2 H = jωµ(σ + jωε)h γ 2 H The propagation constant γ is the square root of γ 2 whose real and imaginary parts are positive. where γ = α + jβ, µε ( ) σ 2 α = ω ωε, and µε ( ) σ 2 β = ω ωε. (33) Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

44 Wave Equations Solution in Cartesian Coordinates The familiar wave equation in one dimension 2 F z = 1 2 F 2 u 2 t 2 has solutions of the form F = f(z ut) and F = g(z + ut) where f and g are arbitrary functions. We can solve for partially conducting media, perfect dielectrics, and good conductors. Normal incidence, Snell s law, standing waves, power, and Poynting vector are important topics. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

45 Wave Equations Poynting Vector The propagation of energy is in the z-direction if the E and H field vectors are in the x-y plane. The direction of energy propagation is given by the Poynting vector. P = E H. (34) This gives the instantaneous rate of energy flow per unit area at a point. P avg = 1 2 Re(E H ). For plane waves, the direction of energy flow is the direction of propagation. Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45

46 Reference Wave Equations Joseph A. Edminister. Schaum s Outline of Theory and Problems of Electromagnetics. McGraw-Hill, Inc., 2nd edition, John D. Kraus, Ronaled J. Marhefka, and Ahmad S. Khan. Antennas for All Applications. Tata-McGraw-Hill, 3rd edition, Nannapaneni Narayana Rao. Elements of Engineering Electromaganetics. Prentice Hall, 4th edition, Ranga Rodrigo (University of Moratuwa) Antennas and Propagation October 20, / 45