Radiation and Antennas

Transcription

1 Chapter 9 Radiation and Antennas. Basic Formulations 2. Hertzian Dipole Antenna 3. Linear Antennas An antenna is a device to transmit or receive electromagnetic power more efficiently with a more directive radiation and a smaller dissipation power. To analyze the radiation pattern of an antenna, current distribution on the antenna is assumed. Then the potentials, the fields, and hence the power flux density are calculated explicitly. The Hertzian dipole where the current distribution is a current element is studied first. To characterize antennas in a quantitative way, the terms of antenna pattern, radiation intensity, directivity, input impedance, and radiation efficiency are introduced. Then linear antennas analyzed. 9.. Basic Formulations If the current distribution over an antenna can be known by measurement or calculation, then the distribution of the vector potential and hence those of the fields can be determined. Since, potential A is given explicitly as Ar) = µ GR)Jr )dv, where Green s function G is given as GR) = e jk R 4πR and k = ω/c is known as the propagation constant in free space. Thereafter, fields B and E can be given explicitly as Br) = Ar) and [ Er) = jωa Φ = jω Ar) + ] Ar), k 2 where the potential continuity equation Lorenz gauge) has been made use of. Note that field E can be determined solely from potential A. In a source-free region J =, field E can be evaluated alternatively in terms of field B as Er) = jωµ ɛ Br), where Ampere s law has been made use of. Thereafter, the time-average power flux density can be determined from the complex Poynting vector P given by Sr) = 2 Er) H r).

2 em Hertzian Dipole Antenna A Hertzian dipole is a very short, thin conducting wire which carries a constant time-harmonic current on it. The Hertzian dipole is the building block of other wire antennas. For wire antennas the current density may described as a line current density. For a z-directed Hertzian dipole located at the origin and of length dl, the line current density J l can be described as ẑi ρ = and z < J l r) = dl 2. elsewhere Then the vector potential A becomes Ar) = µ e jk R 4πR J lr )dl = ẑµ IdlGr). Note that potential A is in the z direction, since the only current element is in that direction. In spherical components, potential A for the z-directed current element located at the origin can be written as Ar) = µ IdlGr)ˆr cos θ ˆθ sin θ). It is soon to be seen that the associated fields are somewhat complicated even for such a simple antenna. A direct expansion leads to the expression of field B as Br) = Ar) = ˆϕ r [ raθ r, θ) r A rr, θ) θ ] = ˆϕjk µ IdlGr) j ) sin θ. k r Note that field B only has the ϕ component which in turn is independent of azimuthal angle ϕ. For evaluating field E, evaluate the divergence A first. Thus Ar) = r 2 r r2 A r ) + = jk µ IdlGr) r sin θ θ A θ sin θ) = µ IdlGr) j ) cos θ. k r jk + r 2 ) cos θ r This divergence is also independent of ϕ. Then evaluate its gradient. Thus Ar) = ˆr r + ˆθ ) A r θ { [ = µ IdlGr) ˆr jk jk + r ) + jk r 2 ] r ) cos θ ˆθ 2 r jk + } r ) sin θ { = kµ 2 IdlGr) ˆr + j 2 k r + 2 ) cos θ + ˆθj kr 2 2 k r + } kr ) sin θ. 2 2 Thereafter, field E is given by Er) = jω A + k 2 A = jωµ IdlGr) { ˆr ) j 2 k r + 2 ) cos θ + kr ˆθ + j 2 2 k r + ) } sin θ. kr 2 2

3 em9 3 It is seen that the formula for electric field is much more complicated than the one for magnetic field. Another way to evaluate field E is to employ Ampere s law in a source-free region. By so doing, one has Er) = = H = jωɛ jωɛ { jωɛ IdlGr) = jωµ IdlGr) ˆr r sin θ sin θh ϕ θ ˆr jk + ) 2 cos θ r r { ) ˆr j 2 k r + 2 k 2 r 2 ˆθ ) rh ϕ r ˆθ r [ jk jk + ) ] sin θ r r 2 ) cos θ + ˆθ + j k r + k 2 r 2 This results agrees with the previous one. In the far zone where k r, the radiated fields become rather simple forms of Er) = ˆθjωµ IdlGr) sin θ Hr) = ˆϕ jk IdlGr) sin θ. Both of these fields are due to A θ and of current origin. It is important to note that in the far zone the magnitudes of the fields decay with /r. Furthermore, the far-zone field components E θ and H ϕ are related in a simple manner as E θ r, θ, ϕ) H ϕ r, θ, ϕ) = ωµ µ = = η, k ɛ where the quantity η 2π Ω is known as the intrinsic impedance of free space. The surface over which the fields have an arbitrary constant phase forms surfaces of spheres of varying radii. Thus the fields due to a Hertzian dipole propagate in the form of spherical wave. In the farzone, the separation distance of two consecutive zero-phase surfaces is a constant and hence can be defined as the wavelength λ of the propagating wave. It is seen that λ = 2π/k. The wave propagates with a speed c and a propagation constant k in the radial r direction. The electric field is perpendicular to the magnetic field; and both fields are perpendicular the propagation direction of the wave. For a localized region, the electromagnetic fields present a plane wave in the far zone. The complex Poynting vector for the Hertzian dipole is then given as Sr) = 2 Er) H r) = ˆrk 2 η I 2 dl 2 sin2 θ r 2. Note that the power radiated in the radial r direction and the power flux density decreases with /r 2 and increases with ω 2. Antenna Pattern The antenna pattern is the radiation pattern of an antenna and depicts the relative farzone field strength versus the ϕ and θ angles at a fixed distance from the antenna. To represent the dependence of field strength as a function of ϕ and θ angles, a three-dimensional plot is more illustrative. However, the plotting can be more easier, if one of the angles is fixed. Antenna pattern at a constant ϕ, that is, the field strength vs. θ is known as the E-plane pattern. And that at θ = π/2, that is, the field strength vs. ϕ is known as the H-plane pattern. } sin θ }.

4 These two patterns are particularly important for most antennas. E-plane pattern function is em9 4 For the Hertzian dipole, the Eθ) = E θ θ) sin θ, ϕ = ϕ where ϕ is arbitrary. And the H-plane pattern function is Eϕ) = E θ ϕ), θ = π/2 which is independent of ϕ. The E- and H-plane patterns can be presented by plotting in polar coordinates in r, θ) or r, ϕ), respectively, where the r coordinate is used to monitor the field strength. Do not confuse this r coordinate in polar coordinates with the radial variable r in spherical coordinates.) By plotting the E-plane pattern for a Hertzian dipole in polar coordinates r, θ) with r = sin θ, the pattern functions for ϕ = and π become a pair of unit circles contacting each other at the origin. This E-plane pattern is strongest at θ = π/2 and vanishes at θ = or π. And, the H-plane pattern is independent of ϕ and the corresponding plotting in polar coordinates becomes a unit circle centered at the origin. Directivity In many cases, it is desired that the radiated power can be concentrated in a particular direction to achieve a higher efficiency. Thus an antenna pattern which is particularly strong in that direction is preferred. In order to represent the distribution of power flux density over various directions, define the radiation intensity Uθ, ϕ) as Uθ, ϕ) = r 2 S av r, θ, ϕ) = r 2 2 Re{Er) H r)}. In this definition, due to the multiplying factor r 2, the radiation intensity U is independent of the distance r. Furthermore, the time-average power flux density is always radially directed, the radiation intensity U is thus defined to be a scalar quantity for compactness. Then the total time-average radiated power P r can be written in terms of the radiation intensity U as P r = Uθ, ϕ) sin θdθdϕ, S where S is a spherical surface of an arbitrary radial distance r, so long as k r. For an arbitrary antenna, the average radiation intensity is given as U av = P r 4π. Since the quantity sin θdθdϕ denotes the solid angle of the surface integration element, the radiation intensity Uθ, ϕ) represents the magnitude of the time-average power flux per unit solid angle. Solid angle is equal to the projected area on the surface of a sphere of radius r divided by r 2 ) The directive gain G D θ, ϕ) of an antenna is the ratio of the radiation intensity in the direction θ, ϕ) to the average radiation intensity and is a dimensionless quantity. That is, G D θ, ϕ) = Uθ, ϕ) U av.

5 em9 5 If an antenna were omnidirectional, its radiation intensity is spherically uniform and the corresponding the directive gain would be unity in any direction. For any practical antenna, the directive gain exceeds unity in some directions and is less than or equal to unity in other directions. The maximum directive gain of an antenna is called the directivity of the antenna and is denoted by D. That is, D = U max U av. Apparently, for an arbitrary antenna, D >. For the Hertzian dipole, the radiation intensity is given as Uθ, ϕ) = r 2 2η E θ r, θ, ϕ) 2 = 32π 2 η ωµ Idl sin θ 2, which is independent of the radial distance r and increases quadratically with angular frequency ω. The total radiated power P r is then given as P r = π ωµ 32π 2 Idl 2 2π sin 3 θdθ. η which is also independent of r and the integral takes the value of 4/3. The directive gain G D is thus given as Uθ, ϕ) G D θ, ϕ) = = 3 U av 2 sin2 θ. The directive gain is strongest at θ = π/2 and vanishes at θ = or π. The directivity of an Hertzian dipole is then D =.5. Radiation Resistance Another factor affects the efficiency of antennas is the dissipation power due to ohmic loss relative to the total radiation power. For a given current the ohmic dissipation power is proportional to the material resistance. A typical value of this resistance is a fraction of Ω. In order to estimate the radiation power, it is helpful to define a parameter known as radiation resistance R r. Thus the time-average radiation power P r can be written analogously as P r = 2 I m 2 R r, where I m is the antenna current at some suitable reference position. Usually, this reference current is taken to be the maximum current along the antenna. For higher radiation efficiency, a higher radiation resistance is usually a desirable feature of an antenna. For the Hertzian dipole, the radiation resistance R r is given as R r = 2P r I m 2 = 2ωµ dl) 2 2πη = η 6π k dl) 2 = 8π 2 ) 2 dl. λ Note that the radiation resistance of the Hertzian dipole increases quadratically with the operating frequency or with the dipole length dl. The radiation resistance tends to be small for a short dipole. As an example, for dl =.λ, R r is as small as.8 Ω. This radiation resistance tends to be much greater than the ohmic dissipation resistance and hence the radiation efficiency is low. Thus it is seen that a wire antenna of a length much shorter than one wavelength is not suitable for an

6 em9 6 efficient radiator of electromagnetic wave. In other words, an antenna is not an efficient radiator at a low frequency, such that its size is much smaller than the corresponding wavelength. Meanwhile, the dissipation resistance R d due to ohmic loss in a wire antenna is given as R d = dl/σs, where S is the transverse area over which current is flowing along the wire. It is known that at high frequencies the current tends to concentrate itself on the surface of a conducting wire with a shallow depth, known as the skin depth δ = 2/ωµ σ), which is usually much less than the wire radius a. Thus, due to the skin effect, the dissipation resistance is given as R d = dl/σδ2πa. Note that P r dl 2 /λ 2, while P d dl/a. Thus P d P r λ2 adl. It is seen that in order to achieve a higher radiation efficiency, a thick and long wire is usually desired. Radiation efficiency As the antenna material dissipates some of the power as heat, the total input power P in is the sum of the radiated power and the ohmic dissipated power P d, as P in = P r + P d. The radiation efficiency ζ r of a lossy antenna is defined as the ratio of the radiated power P r to the input power P in as ζ r = P r = P in + P. d P r For a Hertzian dipole, the radiation efficiency becomes ζ r = / + R d /R r ). For a given R d, the radiation efficiency ζ r increases with increasing radiation resistance R r. In terms of the radiation efficiency, the power gain G p of an antenna is defined as G p = U max P in /4π = Dζ r Linear Antennas Consider a thin z-directed linear antenna of length l which carries a nonuniform current Iz) and is centered at the origin. The magnetic vector potential due to such a linear antenna is also directed in z and is given by the line integral Ar) = ẑµ l/2 l/2 e jk R 4πR Iz )dz, where R = r r r z cos θ. The variation of R in the phase term e jk R ) affects the magnitude and even the sign of the value of Green s function appreciably, especially when k is large enough. By retaining the variation of R only in the phase term and ignoring that in the amplitude term /R), the potential A may be simplified to be Ar) = ẑµ e jk r 4πr l/2 l/2 Iz )e jk z cos θ dz.

7 em9 7 The line integral represents the collective effect of the various current elements distributed over the linear antenna and its actual result will determine the characteristics of the linear antenna. To represent this line integral, define the dimensionless factor F as F θ) = k l/2 sin θ Iz )e jk z cos θ dz, 2I m l/2 where I m is some reference current. By including the factor k /I m, the pattern function F becomes dimensionless. The purpose of including the factor sin θ in the pattern function F is owing to the factor ẑ ˆθ in calculating the field component E θ from a z-directed vector potential A. In terms of the pattern function F, potential A can be rewritten as Ar) = ẑ2i m µ Gr) F θ) k sin θ. The pattern function F is independent of r, while the dependence of potential A on r is incorporated in Gr). Note that in spherical coordinates, = ˆr / r + ˆθ /r θ + ˆϕ /r sin θ ϕ and Gr)/ r = jk j/k r)gr). In the far zone, those terms with /k r can be dropped. Thus, in dealing with an arbitrary scalar function whose dependence on r is of the form Gr), the gradient operator can be approximated to be jk ˆr. Further, as noted in Chapter 2, the derivatives of the base vectors in spherical coordinates with respect to r are all zero. Thus, in dealing with an arbitrary vector function whose dependence on r is of the form Gr), the curl and divergence operators and can be approximated as jk ˆr and jk ˆr, respectively. Thus the del operator can be replaced with jk ˆr. Thereby, the evaluation of fields becomes much simpler. One immediately has and Br) = Ar) = jk ˆr ẑa z = jk ˆϕA z = ˆϕj2I m µ Gr)F θ) Ar) = jk ˆr ẑa z = j2i m µ Gr)F θ) cos θ/ sin θ Ar) = ˆr2I m k µ Gr)F θ) cos θ/ sin θ. Thus, in the far zone, the electric field is given as Er) = jωẑ + ˆr cos θ)2i m µ Gr) F θ) k sin θ = ˆθj2I m η Gr)F θ). Note that this field is contributed from the current, not from the associated charge. Thus F θ) become the E-plane pattern function of a linear dipole antenna. Again, it is seen that E θ r, θ, ϕ) H ϕ r, θ, ϕ) = η. The radiation intensity can be given also in terms of the F function as Uθ) = r 2 2 Re{E θhϕ} = 2η 4π) I m 2 F θ) 2. 2 For the z-directed linear antenna, the radiation intensity is independent ϕ.

8 em9 8 An alternative way to evaluate the fields due to a z-directed linear antenna is to integrate the fields due to a z-directed Hertzian dipole. Thus Er) = ˆθjk l/2 η sin θ l/2 GR)Iz )dz l/2 Hr) = ˆϕ jk sin θ GR)Iz )dz. l/2 Again, by retaining the variation of R only in the phase term and ignoring that in the amplitude term, one will have the same results. The current distribution on an antenna depends on the wiring structure, wire radius, frequency, the feeding, and also on the environments. The determination of the actual current distribution is a rather difficult problem from the viewpoint of field analysis. In some cases, the current distribution can be determined by measuring the magnetic flux linkage over a small loop. However, some information about the current distribution can be inferred. The current distribution for a centerfed linear antenna should be symmetric about the center. Furthermore, the current should vanish at the ends of an opened wire antenna. From theoretical analysis, numerical computation, and experimental measurement, the current distribution on a thin center-fed linear antenna of length 2h is close to the standing wave given as Iz) = I m sin[k h z )]. This current distribution becomes triangular for a short antenna. If the antenna is very short much smaller than one wavelength), then the current vanishes, as expected from the lumpedcircuit concept. The pattern function F is given by the integral F θ) = k h 2 sin θ sin[k h z )]e jk z cos θ dz. h After a lengthy manipulation, it can be shown that the F pattern is given as F θ) = cosk h cos θ) cosk h), sin θ which is a function of dipole length h/λ. When k h, the pattern function F becomes F θ) = { 2 k h cos θ) k h) 2 } sin θ = 2 k2 h 2 sin θ, which increases quadratically with the relative length h/λ. The E-plane radiation pattern F θ) can vary significantly when dipole length exceeds one wavelength. The radiation pattern may become multiple beams. The angles of the maximum radiation intensity and those of the null vary with the antenna length. The radiation pattern in fact is a result due to the interference among the radiation from various current elements on the wire antenna. From the versatile radiation patterns, one can design a linear antenna with the desired characteristics by choosing the antenna length. A computer visualization will be of much help in understanding the radiation characteristics. half-wave dipole antenna

9 em9 9 For a half-wave dipole antenna k h = π/2, the F function becomes F θ) = cos π cos θ) 2, sin θ which has a maximum equal to unity at θ = π/2 and nulls at θ = and π. For a half-wave dipole antenna, the total radiation power P r is given by π P r = 2π Uθ) sin θdθ = η π 4π I m 2 cos 2 π cos θ) 2 dθ = I m 2 W, sin θ where by a numerical integration the integral is found to take a value of.28. The maximum radiation intensity is U max = 2η I m 2 /4π) 2 and the average radiation intensity is U av = P r /4π. Thus the directivity D is then given as D = U max U av = 2η 4π =.64, which is higher than the.5 of the Hertzian dipole. The radiation resistance R r is as high as R r = 2P r I m 2 = 73.Ω. Input Impedance In order to maintain the antenna current which is radiating power outward, an excitation from an external transmitter is needed. A wire antenna needs a gap, usually at the center of the wire, via which the current pumped from an transmitter is impressed into the antenna. Thus an antenna is actually a two-terminal device. Like other two-terminal devices, the input impedance Z in of an antenna is given by the ratio Z in = V in I in, where V in and I in are the input voltage and the input current, respectively. From the Poynting theorem, the total radiation power P r for a lossless antenna is related to the input resistance as P r = Re E J dv 2, where field E is due to the current J induced in the antenna to counteract the electric field due to the excitation. The electric field vanishes in the antenna, except at gap across which the input voltage is V in. Thus the evaluation of the radiation power becomes simpler. That is, { } P r = Re E J dv 2 = Re 2 V iniin = 2 R in I in 2, gap where the input resistance R in is the real part of input impedance Z in. Note that the term Re{E J } <, since the current in a transmitting antenna tends to release power. For a lossless antenna, P in = P r, which states that the input power is totally converted to radiation power. Besides the radiation efficiency ζ r and the directivity D, the input impedance Z in is another factor affecting the efficiency of an antenna. An antenna is usually excited through transmission ant

10 em9 lines. It is known that if the input impedance is not matched to the characteristic impedance of the chosen transmission lines, some of the input power will be reflected back from the antenna as a loss. The value of characteristic impedance depends on the structure of the lines and a typical value is about 5 3 Ω. If the current at the input terminals of a lossless antenna is maximum in magnitude I m = I in ), the radiation resistance R r is just the input resistance R in. Another difficulty with a short antenna is the impedance matching. It has been found that the radiation resistance and hence the input resistance of a short antenna is very small. Furthermore, from a numerical simulation, it can be found that for small antennas the input reactance increases inversely with the length dl. As dl, the input impedance becomes as large as that of an open circuit. Quarter-Wave Monopole Antenna The components of the electric field tangential to the plane bisecting a centered-fed half-wave dipole antenna is evidently zero by reason of symmetry. Thus the length of a half-wave dipole antenna can be cut by one half without affecting its radiation pattern, if one places a quarter-wave antenna vertically over a perfectly conducting ground plane. Conduction current will be induced on the ground plane. By using the method of image, the resultant effect of the current on the planar ground will be the image of the antenna current. The image current has the same direction as the true current in the antenna. In the far zone, the surface current density can be determined from the magnetic field as given by the boundary condition J s = ẑ H. The magnetic field is in the azimuthal ϕ direction. Thus the induced current is in the radial direction. The electromagnetic fields due to the radial current induced on the ground plane should be identical to those due to the vertical image current so far as only the upper half space is considered. The wire end which is close to the ground should be isolated from this ground. Excitation is made between this end and the ground. The total radiated power P r and the radiation resistance R r are half those of the half-wave dipole antenna. In an AM transmitting monopole antenna, the earth ground is used as the ground plane. To improve the conductivity of the ground, radial metal cables buried in the earth are helpful. In mobile radio f =46/44 MHz and λ =2.8/.68 m), the metal forming the body of a car provides the conducting ground for the monopole antenna of length about.52/.7 m). In a hand-held cellular phone f 9/8 MHz, λ 33.3/6.7 cm), the metallic case of the wireless handset provides the conducting ground. The ground plane GP) antenna is suitable for the base station of mobile radio where the monopole antenna is placed on an elevated place, such as mounted on a pole on the roof. In order to make the maximum radiation be in the horizontal plane, the ground plane should have a downward tilt.