( ) 2 ( ) 3 ( ) + 1. cos! t " R / v p 1 ) H =! ˆ" I #l ' $ 2 ' 2 (18.20) * + ! ˆ& "I #l ' $ 2 ' , ( βr << 1. "l ' E! ˆR I 0"l ' cos& + ˆ& 0


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1 Summary Chapter 8. This last chapter treats the problem of antennas and radiation from antennas. We start with the elemental electric dipole and introduce the idea of retardation of potentials and fields an idea stemming from the finite speed of propagation of electromagnetic waves. The fields and properties of antennas follow from these simple terms. Retarded potential is a potential the magnetic vector potential in this case produced by a current at time t Δt and measured at a time t at a distance R. Given a sinusoidal current I cos! t " R / v p we have in the frequency domain: AR = ẑ µi!l R e# j$r Wb m 8. where Δl is the length of a short wire segment carrying the current I and β = ω/v p. This short segment is called a Hertzian dipole antenna. Fields of the short dipole Hertzian dipole see Figure 8.3 H =! ˆ" I #l $ e! j$r sin 4 j$r  A j$r. / m 8. E =! ˆR "I #l $ e! j$r cos j$r j$r 8. 3! ˆ "I #l $ e! j$r sin 4 j$r j$r j$r 3 There are three zones that can be defined near field intermediate field and far field. These zones and their main properties are shown in the following table: definition Approximations Notes Near field Also called: Electrostatic field or Fresnel zone Intermediate field or inductive zone Far field Also called: Radiation field or Fraunhoffer zone βr << or: R <<! / " βr >> or: R >>! / " E! ˆR I "l I cos ˆ "l 3 j#$r j4#$r sin V 3 m 8.7 H! ˆ" I #l 4$ R sin A m 8.5 None. Must use Eqs. 8. and 8. H = ˆ! j"i #l 4$ R e j"r sin E = ˆ! j"#i $l e j"r 4 R A m V m V. / m E is large H small. Terms with / j!r 3 in Eqs. 8. and 8. dominate The fields in Eq. 8. and 8. are correct in all zones since no approximations are used E and H are perpendicular to each other and to direction of propagation. Behavior similar to plane waves but in spherical coordinates Terms with /jβr in Eqs. 8. and 8. dominate Radiation properties of Hertzian dipoles: Time averaged power density: P av = ˆR!I " #l sin 3$ R W m 8.3
2 Radiated power: =!I " #l $!$ #l = I [ W] Radiation resistance =!" 3 #l $ #l = 8" $ [ ] Antenna radiation pattern is the relative normalized strength of the electric or magnetic field intensity field radiation pattern or its power power radiation pattern in the far field. The pattern is usually given as a plot in polar or rectangular coordinates or as a three dimensional representation. E plane plots are radiation patterns in a plane that includes the antenna Hplane plots are in the plane perpendicular to the antenna. Beamwidth is the angle between the two half power points on the radiation pattern. Radiation intensity: U!" = P av R [ W/ sr] 8.46 [ ] 8.48 = U!" /U av =.5 sin! [ dimensionless] 8.5 Average radiation intensity: U av = / 4! W/ sr Antenna directivity: D!" Maximum directivity: d =.5 Antenna power gain: G!" = Antenna radiation efficiency: eff = G!" = = d P in 4#U!" P in R d [ dimensionless] 8.54 [ dimensionless] Magnetic dipole small loop antenna radius<<λ see Fig. 8. Near fields: m # A j#µm H = ˆR cos" ˆ" sin" 4! R 3 $ m 8.75 E =! ˆ" 4$ R sin V m 8.76 Far fields E = ˆ! "µ#m 4$ R e j#r sin V m H =! ˆ" #µ$m 4R e! j$r sin" A m 8.77 where m=πd I [A.m ] is the magnetic dipole moment of the loop with d its radius and I its current. Table 8. summarizes the properties of Hertzian and magnetic dipoles. Arbitrarily long antennas. The properties of arbitrarily long antennas are obtained by viewing them as stacks of Hertzian dipoles. In essence we assume a sinusoidal current along the antenna so that it is zero at its ends assume a differential of length along the antenna assign to it the electric and magnetic fields of the Hertzian dipole Eqs. 8.3 and 8.3 and integrate to find the fields in the far field see Fig The table below summarizes the main properties of arbitrarily long antennas together with the particular but important case of a half wavelength antenna. Arbitrarily long antenna length = L λ/ antenna L = λ/ Iz # # [A] I sin! L $ " z $ 8.79 I cos! z 8.96 " ERθ [V/m] HRθ [A/m] $ cos L / ˆ! j"i cos L / cos! # R e$ jr 8.85 ˆ! ji cos $L / cos " R e# j$r sin # cos $L / ˆ! j"i cos # / cos! # R e$ jr 8.97 ˆ! ji cos " / cos " R e# j$r sin
3 P av Rθ [W/m ] ˆR!I cos #L / cos$ cos #L / 8" R ˆR!I $ cos " / cos# 8" R sin# !I cos #L / cos$ cos #L /.8!I [W] d$ 8. $= 8.9! cos #L / cos$ cos #L /.69! 8. [Ω] " d$ " $= 8.93 D! 4!P av R cos! / cos" 8.4 sin " d D! = " / Depends on length L D! = " / = f e! cos!l / cos" # cos!l / cos! / cos" sin" sin" Note: the integral in the equations above is calculated numerically listed in Table 8. and in Figure 8.5 Monopole antennas. These may be viewed as half dipoles that is a single element perpendicular above a conducting ground. Its properties are identical to the equivalent dipole except that the radiated power and radiation resistance are half that of the equivalent dipole. Also the radiation pattern only exists above ground. Antenna Arrays are multiple antennas in a geometric configuration and driven with specific currents all designed to produce a given radiation pattern. The fields of antenna arrays are obtained by simple superposition of the fields of individual elements Uniform phased array N elements equally spaced on an axis carrying currents of identical magnitude and constant phase difference between two neighboring elements. Broadside array The main beam lobe radiates sideways relative to the axis of the array Endfire array The main beam radiates along the axis of the array. Two element array: see Fig. 8.7 E = ˆ! E f!" R e # j$r e j / cos 8.! = "h sin# cos$ [ rad] 8.9 R is the distance from the first element in the array E and fθφ are the amplitude and radiation pattern of a single element h the separation between elements and ϕ the phase difference. In this case the antennas are parallel to each other and perpendicular to the xaxis. If the antennas are configured differently see Figures 8.9 and 8.3 the results are different because ψ is different see Examples 8. and 8.. Nelement linear array see Fig n E =! E k = ˆ" E f "# e $ jr e j n$ / sin n / R sin / k= V m 8.3 Normalized array factor f n!"# = n sin n$ / sin $ / = n / / sin n h cos" # sin h cos" # 8.34
4 The terms R ψ h ϕ are as for the two element array. Reciprocity and receiving antennas. Reciprocity theorem only applies to isotropic media: If a source is applied to antenna No. and a signal is received in antenna No. then applying the same identical source to antenna No. will produce an identical signal in antenna No.. Effective length is the ratio between the magnitude of the open circuit voltage at the antenna and the electric field intensity in the direction of the antenna Fig Effective aperture: is the ratio of power received by the antenna and the time average power density at the location of the antenna: A ea = P a P!" m # $ 8.38 av For any antenna: A ea = D!" # 4$ m 8.48 Maximum effective area: For Hertzian dipole: A ea max =.5! #$ m 8.49 For λ/ dipole: A ea max =.64! #$ m 8.5 Friis transmission formula is a common design tool in antennas and communication. It defines the power received P rec. in terms of transmitted power P trans. and properties of the transmitting and receiving antennas indicated with indices r and t respectively in lossless media: P rec. = A A er et = D P trans.! R r "#! 4$ D t "#! 4$ ! R Radar and radar crosssection. Radar crosssection is the apparent area of a target based on the power it reflects: P! = R s P #$ m 8.57 i where P s [W/m ] is the power density scattered by the target and P i [W/m ] is the incident power density at the location of the target. Radar equation. This is a modification of the Friis formula for the specific conditions of radar. P r =! A e R 4 # 8.65
5 Long antennas: Assume the current as follows: # # Iz = I sin! L $ " z $ [ A] 8.79 In the far field: E = ˆ! j"i # R e$ jr H = ˆ! ji " R e# j$r $ cos L / cos L / cos! # cos $L / cos $L / cos sin cos #L / P av = ˆR!I cos #L / cos$ 8" R V m A m W / . m =!I =! " cos #L / cos$ cos #L / d$ W $= cos #L / cos$ cos #L / d$ $= [ ] 8.9 [ ] 8.93 D! = P av R = $ cos #L / cos #L / cos! cos #L / cos! $ cos #L /!=" d!!= 8.94 f e! = # cos "L / cos "L / cos! # cos "L / $ cos "L / f p! = cos! 8.95 Half wavelength dipole obtained by setting L=λ/: Iz = I cos! z " [ A] 8.96 E = ˆ! j"i cos # / cos! # R e$ jr H = ˆ! ji cos " / cos " R e# j$r sin P av = ˆR!I $ cos " / cos# 8" R sin# =.8!I In air: = 73.8! D! =.64 cos " / sin! V m A m W  m. / [ W] 8. =.69! " [ ] 8. cos! [#] d = D! = " / =
6 Radiation patterns: f e! = cos " / cos! # f p! = cos " / cos! $ 8.3
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