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简介:本文档为《Antennas and Propagation for Wireless Communicationspdf》,可适用于电信技术领域,主题内容包含ChapterIntroductionAcommunicationslinkorchannelconsistsofasourceofinformat符等。

Chapter 1 Introduction A communications link or channel consists of a source of information, transmitter electronics including modulators, mixers, power amplifiers, or other components, a transmitting antenna or array, the propagation environment, a receiving antenna or array and associated receiver electronics, and signal processing to detect and decode information from the received signal. The purpose of this book is to develop the analytical tools required for an end-to-end model of such a communications link, including antenna and propagation effects as well as signal processing. The key figures of merit for the communications link are the signal to noise ratio (SNR) and channel capacity. Channel capacity is the maximum bit rate that can be reliably sent from transmitter to receiver. For a single communications channel, the capacity is determined by the bandwidth of the channel and the signal to noise ratio (SNR) at the receiver output, which is influenced by the following factors: Transmitter: Total radiated power, antenna radiation pattern, gain, and polarization. Propagation environment: Distance between transmitter and receiver, multipath, blockage, loss, noise, and interference. Receiver: Antenna characteristics, amplifiers and receiver electronics, and signal processing. The goal is to understand each of these aspects of the electromagnetic propagation channel and to model the overall performance of a communications channel in terms of the transmitter characteristics, propagation environment, and receiver system. In order to develop a complete channel model, we must consider antenna theory for both transmitting and receiving antennas, specific antenna types and array antennas, noise theory, propagation channels, and communication theory for both single antennas and multi-antenna systems. This will provide the tools necessary to determine the SNR at the output of a communications link, the channel capacity, and the bit error rate realized with the channel for a specific modulation scheme. These tools will also allow synthesis of a communications system which meets a desired performance criterion. 1.1 Link Budget Analysis A simple tool for propagation analysis is a link budget, which is the relationship between the total transmitted power and the signal power at the receiver output in terms of antenna gain and free space path loss. The link budget, together with the noise level at the receiver output, determines the SNR for a communications system. In this treatment, we will develop models for the various contributions to a system link budget. For complex propagation environments and multiple input multiple output (MIMO) communications systems, a simple link budget analysis is inadequate, and a more sophisticated capacity model must be developed. 1 ECEn 665: Antennas and Propagation for Wireless Communications 2 1.2 Applications Applications of antennas and propagation modeling include everything from a basic point to point mi- crowave communications link or radio broadcast system to modern technologies such as wireless local area networks, satellite uplinks and downlinks, deep space communications, and MIMO systems. Many of the same principles are applicable to other fields beyond voice and data transmission, such as receivers for radio astronomy observations, magnetic resonance imaging, radar, global position systems, and remote sensing. Warnick & Jensen January 3, 2011 Chapter 2 Antennas As a device that transforms a wave on a transmission line to a wave in the space around the antenna, an antenna has two key properties: the input impedance it presents to the transmission line, and the pattern of the radiated fields. Configured as a receiver, the antenna can be modeled as an equivalent voltage or current source connected to a transmission line, with an open circuit voltage or short circuit current induced by an incident field and a given source impedance. Typically, an antenna is modeled as a transmitter, and its receiving properties are inferred using the electromagnetic reciprocity principle. An antenna radiation problem is a boundary value problem, where the fields radiated by the antenna are determined by Maxwell’s equations with material properties given by the shape and composition of the antenna structure and a source excitation connected to the antenna terminals. Maxwell’s equations then determine the electromagnetic fields around the antenna. From these fields, the voltage and current at the antenna terminals can be computed to determine the antenna impedance, and the far fields determine the antenna radiation pattern. 2.1 Antenna Analysis One way to find the fields around the antenna is to use a numerical method to solve the boundary value problem directly. All antenna parameters, including the antenna impedance, can be found in this way. For simple antenna types, it is more convenient to develop approximate formulas for the antenna parameters using analytical techniques. One of the basic analytical techniques of antenna theory is to model the antenna as an equivalent current distribution, which when impressed in free space radiates the same fields as the antenna structure with a given excitation at the terminals. The radiation integral can be used to find the far fields, from which the radiation pattern and radiation resistance can be computed. For a lossless antenna, the radiation resistance is equal to the real part of the antenna impedance. The current is also sometimes used to estimate the additional part of the antenna resistance due to ohmic losses in the antenna structure. A more sophisticated analysis or a numerical method is usually required to model the antenna reactance and obtain an accurate value for the antenna impedance. Analytical current models are typically approximate and can be found only for simple antenna geome- tries. For complex antennas, analytical current models are not available, and numerical methods are used to solve Maxwell’s equations and find the field radiated by the antenna. Common analytical current models and numerical methods used for antenna analysis include: Analytical approximations: Hertzian dipole model (delta function current). Linear antennas: triangular or sinusoidal current distributions. 3 ECEn 665: Antennas and Propagation for Wireless Communications 4 Aperture antennas: aperture field is approximated by the incidence field that illuminates the aperture. Patch antennas: cavity model for fields under the patch. Numerical methods 1D method of moments (MOM) for thin wires. 2D method of moments (MOM) for perfect electric conductor (PEC) objects. 3D method of moments (MOM) for composite dielectric and conducting structures. Finite difference time domain (FDTD) Finite element method (FEM) The analytical approximations provide an equivalent current representation for the antenna, from which the fields radiated by the current source can be found using a Green’s function and the radiation integral. A Green’s function is the field radiated by a point or delta function source for a given set of boundary condi- tions. It can be thought of as the impulse response of space. Boundary conditions may include dielectric interfaces, conductors, and the radiation boundary condition at infinity. The most common case is the free space Green’s function, which is available in analytic form. The field is then given by a radiation integral, which is the convolution of the Green’s function with a current source. 2.1.1 Free Space Green’s Function One approach to finding the fields radiated by a point source is to transform the first order system of Maxwell’s equations into a single second order partial differential equation (PDE). If we do this, we find that the electric field satisfies the PDE [+k2]E(r) = jωµJ(r) (2.1) In homogeneous region with no unbalanced charge, D = 0 together with the identity E + E = 2E can be used in this expression to obtain the Helmholtz equation [2 + k2]E(r) = jωµJ(r) (2.2) To find a Green’s function, we need to solve for E for a point source of the form J(r) = pˆδ(r r′) (2.3) where r′ is the location of the source and pˆ is the polarization. There are difficulties associated with finding the Green’s function for either of these PDEs. The deriva- tive operator in (2.1) is complicated, so it is difficult to solve this equation directly. Equation (2.2) has a simpler derivative operator, but the Helmholtz equation has more solutions than Maxwell’s equations (lon- gitudinal waves), so those nonphysical solutions must be eliminated from the convolution of the Green’s function with the source to obtain a valid electric field. To overcome these difficulties, we can define an auxiliary potential that also satisfies a Helmholtz type PDE from which valid electric and magnetic fields can be derived. Gauss’s law for the magnetic flux density is B = 0 (2.4) Using a theorem from differential geometry, it follows that B is the curl of some vector field, so that B = A (2.5) Warnick & Jensen January 3, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 5 where A is called the magnetic vector potential. Using this in Faraday’s law, (E + jωA) = 0 (2.6) Using another theorem from differential geometry, the quantity in parenthesis must be the gradient of some scalar function, so that E + jωA = φ (2.7) In the static case (ω = 0), φ is the electric potential. Using (2.7) in Ampere’s law leads to [+k2]A = jωµφ µJ (2.8) Using an identity for the Laplacian operator, [2 + k2]A = µJ + jωµφ+ A (2.9) The last two terms on the right are inconvenient, but we can eliminate them. Using the fact that there are many vector fields A that satisfy (2.5) for a given B, we can choose the particular vector potential for which A = jωµφ (2.10) This is known as the Lorenz gauge. Using this in (2.9) leads to the Helmholtz equation [2 + k2]A = µJ (2.11) This is similar in form to (2.2), but all solutions now represent valid electromagnetic fields. Scalar Green’s Function We now need to solve (2.2) for a point source. We will label the solution A(r) for a point source located at r′ as g(r, r′), so that [2 + k2]g(r, r′) = δ(r r′) (2.12) Since free space is homogeneous, we can shift r′ to zero, so that g(r, 0) = g(r) = g(r), and we have [2 + k2]g(r) = δ(r) (2.13) We can simplify the Laplacian considerably since g is now only a function of r. For r > 0, this becomes 1 r2 r ( r2 g r ) + k2g(r) = 0 (2.14) If we let u(r) = rg(r), then d2 dr2 u(r) + k2u(r) = 0 (2.15) which has the general solution u(r) = Aejkr +Bejkr (2.16) The radiation boundary condition at infinity implies that waves must be outgoing, so we must have B = 0, and g(r) = A ejkr r (2.17) Warnick & Jensen January 3, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 6 It now remains to find the constant A. We will do this by ensuring that the left-hand side of (2.13) integrates to 1 over a volume containing the origin. Integrating both sides of (2.13) over a ball V of radius r leads to V [2 + k2]Ae jkr r dr = 1 (2.18) If the radius of V is small, then this becomes V [2 + k2]Ae jkr r dr ' V 2A1 r dr = A V 1 r dr = A S 1 r dS = A S 1 r2 r2 sin θ dθ dφ = 4piA from which we have A = 1/(4pi). Shifting the source point from the origin back to r′ leads to the final result g(r, r′) = ejk|rr′| 4pi|r r′| (2.19) This is the scalar free space Green’s function. 2.1.2 Radiation Integral The magnetic vector potential for an arbitrary source distribution J is given by the integral of the scalar Green’s function weighted by the source distribution. Physically, we are using the linearity of the problem to add up the fields radiated by many small point sources that combine to make up the source distribution. This leads to the radiation integral A(r) = µ g(r, r′)J(r′) dr′ (2.20) Since free space is a shift-invariant medium, the Green’s function can be written in the form g(rr′), which places the radiation integral into a convolution form. The electric field can be found in terms of the magnetic vector potential using (2.7) and the Lorenz gauge, so that E = jωAφ = jωA+ 1 jωµ A (2.21) = jω [ 1 + 1 k2 ] A Inserting (2.20) for the vector potential, E(r) = jωµ [ 1 + 1 k2 ] g(r, r′)J(r′) dr′ (2.22) This is the free space radiation integral for the electric field in terms of the scalar Green’s function and the electric current density. Warnick & Jensen January 3, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 7 2.1.3 Far Field Approximation Often in antenna analysis we are only interested in the fields far from the antenna. In this case, we can simplify the radiation integral considerably. If the source is near the origin and the field observation point r is far from the origin, then |r r′| = (x x′)2 + (y y′)2 + (x z′)2 ' x2 2xx′ + y2 2yy′ + z2 2zz′ = r2 2r r′ = r 1 2rˆ r′/r ' r(1 rˆ r′/r) = r rˆ r′ Using this in (2.19), g(r, r′) ' e jkr 4pir ejkrˆr ′ (2.23) To first order in 1/r, the derivative operators in (2.22) can be approximated using e jkr r = rˆ r ejkr r +O(1/r2) = rˆ ( jk e jkr r e jkr r2 ) +O(1/r2) ' jkrˆ e jkr r so that ' jkrˆ far from the source. Using these results in (2.22) leads to the far field radiation integral E(r) = jωµ(1 rˆrˆ)e jkr 4pir ejkrˆr ′ J(r′) dr′ (2.24) The rˆrˆ term subtracts out waves with electric field in the r direction, since these are longitudinal waves and are not valid solutions of Maxwell’s equation. The identity term is precisely what we would have obtained if we had solved (2.2) directly. The integral N(r) = ejkrˆr ′ J(r′) dr′ (2.25) is much like a Fourier transform of the source. This quantity is called the vector current moment. In terms of N , E = jωµ(1 rˆrˆ)e jkr 4pir (θˆNθ + φˆNφ + rˆNr) = jωµe jkr 4pir (θˆNθ + φˆNφ) (2.26) When evaluating the integral over r′, if the source matches a rectangular geometry, it is often convenient to use spherical coordinates for r and rectangular coordinates for r′, so that rˆ r′ = (xˆ sin θ cosφ+ yˆ sin θ sinφ+ zˆ cos θ) (x′xˆ+ y′xˆ+ z′zˆ) = x′ sin θ cosφ+ y′ sin θ sinφ+ z′ cos θ (2.27) Warnick & Jensen January 3, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 8 2.1.4 Magnetic Current Often in antenna analysis it is convenient to use equivalent magnetic currents in addition to electric currents. There are apparently no isolated magnetic charges in nature, but fictitious magnetic currents can still be introduced mathematically into Faraday’s law by adding a source term M : E = jωB M (2.28) If there are no unbalanced electric charges, then D = 0, and therefore we can represent D as a curl according to D = F (2.29) Following a derivation similar to that of (2.11), a Helmholtz equation for F can be obtained, which leads to the Green’s function solution E(r) = g(r, r′)M(r′) dr′ (2.30) In the far field limit, E(r) ' jk e jkr 4pir rˆ L(r) (2.31) where L(r) = ejkrˆr ′ M(r′) dr′ (2.32) 2.1.5 Electric and Magnetic Currents Combining (2.26) and (2.31) leads to the complete far field radiation integral for electric and magnetic currents, E(r) ' jk e jkr 4pir [ θˆ(ηNθ Lφ) + φˆ(ηNφ + Lθ) ] (2.33) The magnetic field is H(r) ' jk e jkr 4pir [ θˆ(Nφ Lθ/η) + φˆ(Nθ Lφ/η) ] (2.34) These expressions represent the most general possible form for a spherical wave, which decays in amplitude as 1/r and has spherical phase fronts (r = constant). It can be seen by inspecting these two expressions that the electric and magnetic far fields are orthogonal and have no longitudinal component in the rˆ direction. Warnick & Jensen January 3, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 9 2.2 Antenna Parameters An antenna is a transformer between a transmission line and free space. To describe an antenna, we must characterize its properties as a transmission line load (input impedance) and the distribution of the electro- magnetic energy that it radiates into space (radiation pattern). There are a number of key parameters and concepts that we can use to describe antenna properties. We will first consider antennas as transmitters, and then we will use the reciprocity theorem to determine the receiving properties of an antenna. 2.2.1 Antenna Parameter Definitions Radiation pattern: An antenna radiation pattern is the angular distribution of the power radiated by an antenna. If an antenna radiates fields E(r, θ, φ) and H(r, θ, φ), then the time average far field power density radiated at the point r has the form Sav(r, θ, φ) = 12Re [ E(r)H(r)] = |E(r)|2 2η rˆ ' f(θ, φ) 1 r2 rˆ, r (2.35) where we have lumped all the angle dependence into f(θ, φ). This angular dependence is the radiation pattern of the antenna. It is customary to normalize the radiation pattern to a maximum value of unity, so that the radiation pattern is defined to be f(θ, φ)/fmax. While the power density pattern is most important, one can also look at the field intensity, phase, or polarization patterns of an antenna. Isotropic pattern: Equal power is radiated in all directions, so that f(θ, φ) = 1. No real antenna has this pattern, but the isotropic radiator is important as a reference pattern with which to compare other antenna radiation patterns in order to define directivity. Omnidirectional pattern: f(θ, φ) = f(θ), so the pattern is independent of azimuthal angle. Pattern cut: In general, a radiation pattern is a two-dimensional function which requires a 3D plot to visualize. For convenience, it is common to plot a slice of the pattern, such as a theta cut, f(θ, φ0) with φ0 fixed. E-plane cut: Pattern in the plane containing the electric field vector radiated by the antenna and the direction of maximum radiation. H-plane cut: Pattern in the plane containing the magnetic field vector radiated by the antenna and the direction of maximum radiation. Pattern lobes: Local maxima in the radiation pattern. The main lobe is the lobe which reaches fmax. Sidelobes are smaller lobes. A back lobe is a pattern lobe near the opposite direction of the main lobe. Sidelobe level: Ratio of fmax to the peak value of the largest sidelobe. Beamwidth: There are several ways to specify the angular width of the main lobe. The most common are the half-power beamwidth (HPBW) and null to null beamwidth. The half-power half beamwidth (HPBW/2) is also used. Pencil beam pattern: Radiation pattern with a small main beamwidth. Warnick & Jensen January 11, 2011 ECEn 665: Antennas and Propagation for Wireless Communications 10 Directivity: Ratio of radiated power density in a given direction to the power density of an isotropic reference antenna radiating the same total power. The total radiated power is Prad = S Sav dS (2.36) where S is a closed surface containing the antenna. The power density radiated by an isotropic antenna is Siso = rˆ Prad 4pir2 (2.37) By the definition, the directivity pattern is D(θ, φ) = Sav(r) Prad/(4pir2) (2.38) The directivity D is the maximum value of the directivity pattern. Directivity and beamwidth are inversely related. Partial directivity: In many cases, the receiver does not capture all the power radiated by the transmit- ting antenna, but instead only the power in one polarization pˆ. Partial directivity is defined to be the radiation intensity corresponding to a given polarization divided by the total radiated power averaged over all directions [IEEE Standard 145-1993]. We can modify (2.38) to be the partial directivity with respect to the polarization pˆ by replacing the radiated power density with Sav,p = |pˆ E(r)|2 2η (2.39) in the definition (2.38) of directivity.

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