Unit 2: Small Antennas + Some Antenna Parameters

Transcription

1 Unit 2: Small Antennas + Some Antenna Parameters Antenna Theory ENGI 9816 Khalid El-Darymli, Ph.D., EIT Dept. of Electrical and Computer Engineering Faculty of Engineering and Applied Science Memorial University of Newfoundland St. John's, Newfoundland, Canada Winter 2017 K. El-Darymli Winter /125 Outline 1 The Fundamental Source of Radiated Energy 2 The innitesimal (a.k.a. Hertzian) Dipole Reality and Useful Fiction The Vector Potential for a Uniform-Current Element The Electric and Magnetic Far-Fields for the Innitesimal Dipole Power Density and Radiated Power Near and Intermediate Fields of the Hertzian Dipole Near-Field (kr << 1) Intermediate-Field (kr > 1) 3 General Antenna Parameters and Characteristics Radiation Pattern and Related Concepts Power and Field Patterns, Polarization, and Principal Patterns Classes of Radiators Radiation Power Density, Radiated Power, and Radiation Resistance Radiation Intensity, Directivity and Gain Half-Power Beamwidth Reciprocity More on Reciprocity 4 A Small Current Loop 5 Ackgt K. El-Darymli Winter /125

2 We have seen from Maxwell's equations that a time-varying B eld (or H eld) implies a time-varying E eld (or D eld) and vice versa. In Term 6 it was observed that one solution to these equations represented (plane) wave energy propagating through some space (free or otherwise). We know from Term 6 that steady (dc) currents produce steady magnetic elds. It should be obvious, then, that time-varying currents will produce time-varying H elds and, thus, also time-varying E elds. Clearly, a time-varying current necessitates an accelerating charge. Therefore, to have e-m energy radiating away from a source, the source must consist of accelerating charge(s) a stationary charge or one moving with a constant velocity does not radiate. There are several simple ways in which a charge may accelerate and thus radiate: 1 charge may oscillate with simple harmonic motion along a straight wire; 2 charge may move with a constant speed along a bent or curved wire; 3 charge may encounter a termination (load), a truncation (wire not innite), or a discontinuity (e.g., going from one wire to another of dierent size or electrical parameters). In these cases, electric charge may be caused to change direction i.e., reect. Many such possibilities exist wanted and unwanted. K. El-Darymli Winter /125 Illustration (from Balanis) K. El-Darymli Winter /125

3 To quantify this idea, consider a packet of electric charge moving along a straight thin conductor in the z-direction, say (illustration is from Balanis). Let ρ L linear charge density in C/m and v z velocity in the ẑ direction. Then the current, I, is given by I = dq dt where Q is charge and t is time. K. El-Darymli Winter /125 Since dq = d(ρ L z), and if ρ L is constant in time so that dq = ρ L dz, the current may be written as I = ρ Ldz dt Taking the derivative w.r.t. time, where a z is acceleration. = ρ L v z. di dt = ρ dv z L dt = ρ La z. For a pulse of length l, we may multiply this result by l to get l di dt = lρ La z or using overdot for the time derivative li = Q v z Q = lρ L is the charge in the pulse of length l. For radiation to occur, this equation must hold i.e., current must have a non-zero time derivative (İ 0 or Q 0) charges must accelerate. K. El-Darymli Winter /125

4 Reality and Useful Fiction In Term 6, it was briey noted that a two-wire open-circuit transmission line could be congured as a dipole antenna. Illustration (from Balanis): For l = λ/2, the current, I, has a node at each end of the wire, the amplitude decreasing continually away from the centre. The wavelength of I is λ. For other lengths, the current pattern may be more complicated; however, current nulls or nodes still occur at the wire ends for the innitesimally thin dipole. K. El-Darymli Winter /125 Reality and Useful Fiction There exist methods of making the current more uniform. Consider, rst, a l = λ/4 monopole antenna which is fed against an ideal ground plane as shown: As we'll discuss in more detail later, there is a λ/4 reection in the perfect ground. This monopole will resonate at the same frequency and provide the same pattern shape as a λ/2 dipole (more later on this). Notice that there is still a current null at the top of the antenna. K. El-Darymli Winter /125

5 Reality and Useful Fiction The shape of the current pattern may be modied electrically by shortening the antenna somewhat and base-loading it with an inductance to provide the same resonance. Across the inductor section (see illustration below), the voltage is essentially linear and the current uniform, making for a more uniform overall current loop. λ/4 monopoles may not be a good idea when f < several megahertz (e.g., at f = 1MHz, λ/4 = c/f = 300 = 75m. 4 4 K. El-Darymli Winter /125 Reality and Useful Fiction A short wire antenna has a triangular current distribution, since the current itself has to reach a null at the end the wires. Hence, another method to make the current approximately more uniform is by adding capacitor plates. The small capacitor plate antenna is equivalent to a Hertzian dipole and the radiated elds can also be described by using the results of the innitesimal antenna. K. El-Darymli Winter /125

6 Reality and Useful Fiction The innitesimal antenna is a suitable model to study the behavior of the elementary radiating element called Hertzian dipole. Consider two small charge reservoirs, separated by a distance l λ, which exchange mobile charge in the form of an oscillatory current. The Hertzian dipole can be used as an elementary model for many natural charge oscillation phenomena. The radiated elds can be described by using the results of the innitesimal antenna. K. El-Darymli Winter /125 Reality and Useful Fiction With the idea of a uniform current distribution in mind, we are next going to consider an innitesimal dipole (l λ ) carrying such a distribution and oriented as shown: Letting I 0 be the current magnitude (and it may be complex), I (z ) = I 0 ẑ (phasor domain) (2.1) K. El-Darymli Winter /125

7 Reality and Useful Fiction You may visualize how such a structure might be used in a repeated way to give an extended antenna element. I is constant over a dierential element but may vary from one element to the next over the length of the wire. Thus, this innitesimal dipole idea, while ctitious in nature, will be very useful in building up to a real situation. K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Consider again the innitesimal, uniform-current, dipole sometimes referred to as a Hertzian dipole oriented as previously: Note that x = 0 and y = 0. We wish to determine the vector potential with a view to obtaining the E and H elds. K. El-Darymli Winter /125

8 The Vector Potential for a Uniform-Current Element In equation (1.40) for the vector potential, we replace J( r )dv with I (z )dz ẑ since the source is oriented along the z-axis and located as shown. Thus, the volume integral reduces to a line integral and since (2.2) Using I (z ) from equation (2.1), A( r) = ẑ µi 0 4π l/2 l/2 e jk x 2 +y 2 +(z z ) 2 x 2 + y 2 + (z z ) 2 dz (2.3) While the integration in (2.3) is not available in closed form (and thus neither are the E and H elds), it does yield to some useful approximations. CAUTION: In antenna theory, very little of the often formidable analysis can be carried out exactly i.e., in closed form. K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Before considering (2.3) further, we note the following terms associated with elds radiated by antennas: 1 The Reactive Near-Field Region is that portion of the near-eld region immediately surrounding the antenna wherein the reactive eld predominates. The energy in this eld does not radiate away from the antenna but rather oscillates near the antenna like in a resonator. It is trapped near the antenna. 2 The Radiating Near-Field (or Fresnel Zone) is that region of the antenna eld between the reactive near-eld and the far-eld where radiation (rather than reactive) elds are predominate and wherein the angular eld distribution is dependent on the distance from the antenna. If the antenna's maximum dimension is much smaller than a wavelength, this region may not exist. 3 The Far-Field (or Fraunhofer) Region is that region of the radiation antenna eld where the eld distribution is essentially independent of the distance from the antenna. This is the zone in which we will be mainly interested in this course. K. El-Darymli Winter /125

9 The Vector Potential for a Uniform-Current Element The distances broadly dividing these regions are depicted below for l > λ: where: l antenna length (or its largest dimension) λ wavelength of the radiation Note: For small dipole antennas The Fresnel region is dened such that:, where K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Evolution of pattern from near to far eld (taken from Balanis): To be valid, l must also be large compared to the wavelength (l > λ ). K. El-Darymli Winter /125

10 The Vector Potential for a Uniform-Current Element Approximation of A( r) Now we return to the vector potential of equation (2.3) and seek to determine a good approximate form. x 2 + y 2 + (z z ) 2 = ; ; since r z for the Hertzian dipole as long as P is not on the antenna. K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Approximation of A( r), cont'd Recall the binomial series: Here, x = 2z cosθ r Hence, r 1 2z cosθ r = Therefore, for the Hertzian dipole, x 2 + y 2 + (z z ) 2 r z cosθ. (2.4) Using (2.4) in (2.3) gives A( r) ẑ µi 0 4π l/2 l/2 e jk(r z cosθ) r z cosθ dz. K. El-Darymli Winter /125

11 The Vector Potential for a Uniform-Current Element Approximation of A( r), cont'd Again, because r z, in the amplitude portion of the integrand, we may use r z cosθ r to get A( r) ẑ µi 0 4π l/2 e jk(r z cosθ) dz. l/2 r Notice that the last approximation has not been used in the phase expression WHY NOT? At a particular position dictated by r, we have µi l/2 0 A( r) ẑ 4πr l/2 e jkr e jkz cosθ dz Note:, hence, K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Approximation of A( r), cont'd On multiplying by ( 2l 2l ) : ( ) [( ) ] µi0 l kl A( r) ẑ e jkr Sa cosθ 4πr 2 (2.5) Illustration: Here Sa is the familiar sampling function Sa(x)= sinx x. Recall, Aside: (a denition of the Dirac delta function). K. El-Darymli Winter /125

12 The Vector Potential for a Uniform-Current Element Approximation of A( r), cont'd Since we are dealing with the innitesimal dipole it seems reasonable to further consider the situation kl 2 cosθ 1 or even kl 2 1 kl 2 2πl λ 2 l λ/π From the note on the sampling function, in this case, ( ) kl Sa 2 cosθ 1 (2.6) To use the approximation to the innitesimal dipole, a rule of thumb stipulates that the overall antenna length must be very small usually, l 0.02λ. Also, remember that for this analysis I (z ) is a constant. K. El-Darymli Winter /125 The Vector Potential for a Uniform-Current Element Approximation of A( r), cont'd Using approximation (2.6), equation (2.5) becomes for the vector potential of the innitesimal dipole may be approximated as ( ) µi0 l A( r) ẑ e jkr (2.7) 4πr where e jkr r is referred to as the spherical propagation factor. The governing assumptions are that l λ and l r, the last assumption being ensured by our stipulation that r z. The form of equation (2.7) suggests a spherical eld a wave travelling radially outward if we assume a e jωt time dependency. Again, it must be emphasized that this is the Hertzian dipole result. K. El-Darymli Winter /125

13 The Electric and Magnetic Far-Fields for the Innitesimal Dipole It is easily deduced from Section 1.3 (Do This) that (2.8) Note too, for the far-eld, E may be calculated from by setting since there is no source in the far eld. ; Hence, K. El-Darymli Winter /125 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Hint to get the elds: Convert A to spherical coordinates before taking the curl (in spherical coordinates). Also, In the far eld,. If you get H rst, higher order terms can't be neglected to the general form of E in (2.10) and (2.11). K. El-Darymli Winter /125

14 The Electric and Magnetic Far-Fields for the Innitesimal Dipole and we have also, by denition, that B = A or H = 1 µ A. (2.9) E and H may be found (TRY THIS) in spherical coordinates (using (2.7)(2.9) and the appropriate coordinate transformations) to be given by E r = η I 0lcosθ 2πr 2 E θ = jη ki 0lsinθ 4πr [ ] e jkr (2.10) jkr [ jkr 1 (kr) 2 ] e jkr (2.11) E φ = 0 (2.12) H r = H θ = 0 (2.13) H φ = j ki [ 0lsinθ ] e jkr 4πr jkr (2.14) K. El-Darymli Winter /125 The Electric and Magnetic Far-Fields for the Innitesimal Dipole The Far-Field E Consider, rst, E θ. As long as k is non-negligible, we may stipulate the far eld by r 1. Then the 1/r 2 and 1/r 3 terms in (2.11) will be negligible in comparison to the 1/r term. Therefore, in the far-eld E θ jηki 0lsinθ 4πr e jkr. (2.15) Furthermore, E r E θ since the former contains only 1/r 2 and 1/r 3 terms. Since E φ = 0, we see that the E θ eld dominates the E-eld radiation eld of the innitesimal dipole. To emphasize this, we write that E = Erˆr + E θ ˆθ + E φ ˆφ K. El-Darymli Winter /125

15 The Electric and Magnetic Far-Fields for the Innitesimal Dipole The Far-Field E, cont'd to a good approximation becomes E Eθ ˆθ or, for future reference E [ jηki0 le jkr 4πr ] sin θ ˆθ (2.16) K. El-Darymli Winter /125 The Electric and Magnetic Far-Fields for the Innitesimal Dipole The Far-Field H From equations (2.13) and (2.14), H = Hrˆr + H θ ˆθ + H φ ˆφ = H φ ˆφ. Using r 1 we have H [ jki0 le jkr 4πr ] sin θ ˆφ (2.17) K. El-Darymli Winter /125

16 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Observations 1. Both E and H have a sinθ dependency, indicating the presence of maxima and minima in the eld of observation (no spherical symmetry as for the vector potential A). 2. There is no φ dependency in either E or H. 3. E H is in the ˆθ ˆφ = ˆr direction, indicating a wave whose E and H are in the direction of propagation and are mutually perpendicular (recall the Poynting vector). In Term 6, we called this a transverse electromagnetic wave (TEM wave). 4. From observations (1) and (2) we see that the far-eld E and H space patterns in the vertical plane are doughnut-shaped. K. El-Darymli Winter /125 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Observations, cont'd A helpful applet: Show that for the vertical plane (i.e., φ=constant plane) the E θ and H φ patterns are circular. K. El-Darymli Winter /125

17 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Observations, cont'd 5. The space containing the dipole is unbounded and homogeneous. The intrinsic impedance of the space from (2.16) and (2.17) is given by η = E θ Hφ. If we have the current element in free space, of course, µ = µ 0 and ε = ε 0 and η = η 0 120π. 6. It may be argued from (2.16) or (2.17) that at a particular instant in time an observer at r sees what was happening at the antenna r/c seconds earlier. Consider, for example, (2.17). Recalling that k = ω/c (assuming free space), equation (2.17) may be cast as [ ] jω I 0 le jω r c H sinθ ˆφ ; I o = I o (ω) c 4πr K. El-Darymli Winter /125 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Observations, cont'd We note jω/c transforms to (1/c) / t in the time domain; If F (ω) f (t) is a Fourier transform pair, then F (ω)e jωa f (t a) where, here, a = r/c. K. El-Darymli Winter /125

18 The Electric and Magnetic Far-Fields for the Innitesimal Dipole Observations, cont'd Thus, noting that I 0 is a phasor in the above expression for H (which is really H( r,ω)) transforms in the temporal sense to H( r,t) 1 c t [ ( ) ] i t r c l sinθ ˆφ 4πr where i(t) is the time domain current. As expected, the H eld observed at r at time t depends on the value of the current at ( t r c ). It must be emphasized that the results of the preceding analysis are useful for very short (in terms of wavelength) antennas provided that the current is uniform. K. El-Darymli Winter /125 Power Density and Radiated Power Power Density We now consider the nature of the power associated with the radiation elds calculated in (2.16) and (2.17). Initially, we note that in the time domain (i.e., time-harmonic elds) Similarly, E and H being the phasor elds given by (2.16) and (2.17). Recall from Term 6 that the Poynting vector, P, is given by P = Ẽ H (2.18) and is the power per unit area (i.e., power density) directed perpendicular to both Ẽ and H. K. El-Darymli Winter /125

19 Power Density and Radiated Power That is, P is in the direction of propagation for the far eld region. Recall: P = Ẽ H = Hence, P = = = K. El-Darymli Winter /125 Power Density and Radiated Power Therefore, P = 1 2 Re { E H } Re { E H e j 2ωt } (2.19) Here, E and H are themselves phasors not time dependent. Thus the rst term in (2.19) is time-independent, while the second term is time-harmonic. Since the time-average of a time-harmonic quantity is zero, the time-averaged Poynting vector is given simply as P a = 1 2 Re { E H } (2.20) The second term is simply an oscillatory (reactive) power density with radian frequency 2ω. K. El-Darymli Winter /125

20 Power Density and Radiated Power We note in passing that if the complete expressions given by (2.11) and (2.14) were used for the elds, the time-averaged radiated power density would still be given by (2.20). Carrying out (2.20) using (2.16) and (2.17) we have This is the time-averaged (real) radiated power density of the current element. Note that I 0 2 = I 0 I 0 for a complex current I 0. In general then P a = ηk2 I 0 2 l 2 32π 2 r 2 sin 2 θˆr = 1 2η E 2ˆr (2.21) K. El-Darymli Winter /125 Power Density and Radiated Power This is the radiation power density in any ˆr direction. Therefore, the 3-d shape of P a is built up by rotating sin 2 θ in the z-y plane (for example or any other constant φ plane) about the z-axis. Illustration: 2-D power pattern. 3-D power pattern (simulated in FEKO). K. El-Darymli Winter /125

21 Power Density and Radiated Power Power Radiated (P r ) Consider a spherical surface of radius r enclosing the current element: Helpful applets: Vector dierential surface area is: d S r = ds r ˆr = r 2 sinθdθdφ ˆr. Also, 0 θ π and 0 φ 2π. K. El-Darymli Winter /125 Power Density and Radiated Power Since P a in (2.21) is the power per unit area, P r is obtained by a simple integration of P a over the spherical surface. P r = = = ASIDE: K. El-Darymli Winter /125

22 Power Density and Radiated Power Thus, since the remaining integral has a value of 4/3, in watts. P r = ηk2 I 0 2 l 2 12π (2.22) This analysis is ne for short antennas (i.e., short compared to a wavelength) with uniform current distribution. (A rule of thumb is l λ/50). K. El-Darymli Winter /125 Near and Intermediate Fields of the Hertzian Dipole Under the assumptions of a Hertzian Dipole (l << λ, radius << λ) we have shown that the elds may be approximated as E r = η I 0lcosθ 2πr 2 E θ = jη ki 0lsinθ 4πr [ ] e jkr (2.23) jkr [ jkr 1 (kr) 2 ] e jkr (2.24) E φ = 0 (2.25) H r = H θ = 0 (2.26) H φ = j ki [ 0lsinθ ] e jkr 4πr jkr (2.27) It is customary to dene the nature of the elds in terms of radian distance r = λ/2π from the source. Clearly, the radian distance implies kr = 1. The near eld is dened as that region for which kr << 1, while the intermediate eld requires kr > 1 and in the far eld kr >> 1. Read Section of the text, pp K. El-Darymli Winter /125

23 Near and Intermediate Fields of the Hertzian Dipole Near-Field (kr << 1) Examining equation (2.23), it is clear that for the near-eld condition, the dominant term in E r is that involving 1 jkr. Thus, we approximate E r j ηi 0le jkr 2πkr 3 cosθ (2.28) Similarly, the remaining elds may be approximated as E θ j ηi 0le jkr 4πkr 3 sinθ (2.29) E φ = 0 ; H r = H θ = 0 (2.30) H φ I 0le jkr 4πr 2 sinθ (2.31) K. El-Darymli Winter /125 Near and Intermediate Fields of the Hertzian Dipole Near-Field (kr << 1), cont'd It may be immediately observed that the E-eld and H-eld are in phase quadrature and thus no time-average power will result for this near eld. This is easily veried by showing (DO THIS) P a = 1 2 Re { E H } = 0 (2.32) K. El-Darymli Winter /125

24 Near and Intermediate Fields of the Hertzian Dipole Near-Field (kr << 1) Notice that this does not say that kr >> 1. An examination of equation (2.23) reveals the rst term in square brackets to be dominant for kr > 1 and E r ηi 0le jkr 2πr 2 cosθ (2.33) Similarly, the remaining elds may be approximated as E θ jηki 0le jkr 4πr sin θ (2.34) E φ = 0 ; H r = H θ = 0 (2.35) H φ j ki 0le jkr 4πr sin θ (2.36) K. El-Darymli Winter /125 Near and Intermediate Fields of the Hertzian Dipole From equations (2.34) and (2.36), we observe that there is now a time-average radial power ow, but from (2.33) and (2.36) a reactive component is also evident. This is easily established in the usual way: P a = 1 2 Re { E H } = 1 ) {(E 2 Re r ˆr + E θ ˆθ Hφ ˆφ } P a = 1 2 Re { E r H φ ˆθ + E θ H φˆr }. (2.37) Noting that and that E r H φ = jηk I 0 2 l 2 8π 2 r 3 sinθ cosθ E θ H φ = ηk2 I 0 2 l 2 16π 2 r 2 sin 2 θ, K. El-Darymli Winter /125

25 Near and Intermediate Fields of the Hertzian Dipole it is obvious that for the intermediate eld P a = 1 2 E θ Hφˆr (2.38) while the ˆθ component is reactive (is it inductive or capacitive or a combination?) Exercise: Show that the E-eld vector for the intermediate eld traces out an ellipse (in time), the plane of rotation being parallel to the propagation direction. (This is referred to as the cross eld.) K. El-Darymli Winter /125 General Antenna Parameters and Characteristics Having examined the far-eld characteristics of an innitesimal dipole, we now consider some (in no way an exhaustive list) features common to all antennas. For now, whenever possible, we'll relate them to the foregoing small-current-element analysis. In fact, we have seen some of the following already. The denitions may be found in the IEEE Standard for Denitions of Terms for Antennas, in IEEE Std (Revision of IEEE Std ), pp.1-50, March , URL: K. El-Darymli Winter /125

26 Radiation Pattern and Related Concepts An antenna radiation pattern or simply antenna pattern is dened as a mathematical function or graphical representation of the radiation properties of the antenna as a function of the space coordinates. In most cases, this pattern is determined in the far-eld context this is almost exclusively the case in this course. The term radiation properties includes power ux density, radiation intensity, eld strength, directivity, phase and polarization. We will consider some of these. K. El-Darymli Winter /125 Radiation Pattern and Related Concepts A suitable coordinate system with some terminology is shown below (taken from Balanis). NOTE: â r = ˆr, â θ = ˆθ, â φ = ˆφ K. El-Darymli Winter /125

27 Radiation Pattern and Related Concepts (i) Power and Field Patterns A trace of the received power at a constant radius from the antenna is called the power pattern. A similar trace of the magnetic (or electric) eld along a constant radius is called an amplitude eld pattern. In Sections and 2-2-4, we alluded to these for the innitesimal dipole. The radiation patterns for this and more general structures are shown below (taken from Balanis). Pattern for innitesimal dipole (omnidirectional pattern but not isotropic): K. El-Darymli Winter /125 Radiation Pattern and Related Concepts General Power Pattern Polar Pattern: various radiation lobes and the rst null and half-power beamwidths of a generic antenna pattern. K. El-Darymli Winter /125

28 Radiation Pattern and Related Concepts General Power Pattern, cont'd Linear Pattern: a linear plot of the radiation intensity versus θ. K. El-Darymli Winter /125 Radiation Pattern and Related Concepts (ii) Polarization There are several aspects to the notion of polarization in e-m theory. First, we dene the polarization of a radiated wave as that property of an e-m wave describing the time-varying direction and relative magnitude of the electric eld vector. Specically, at a xed position in space, the locus of the tip of the Ẽ-eld vector and the sense in which this gure is traced as observed along the direction of propagation dene the polarization of the wave. The general polarization is elliptical, which degenerates in important special cases to linear or circular polarization. The elliptical or circular polarizations may be left-handed or right-handed, depending on the hand for which the curl of the ngers is in the direction of rotation of the Ẽ-eld vector when the thumb points in the direction of propagation. Helpful Applets/Animations: K. El-Darymli Winter /125

29 Radiation Pattern and Related Concepts (ii) Polarization, cont'd A simplied sketch (taken from Applied Electromagnetism, 3 rd edition, by L.C. Shen and J.A. Kong, PWS Publishing Company, Boston, 1995) is shown below. Polarization of an electric eld: (a) Linear polarization. (b) Circular polarization (left-hand circular polarization if wave propagates in the ẑ direction. (c) Circular polarization (right-hand if wave propagates in ẑ direction). (d) Elliptical polarization. K. El-Darymli Winter /125 Radiation Pattern and Related Concepts (ii) Polarization, cont'd Polarization of an antenna in a given direction is dened as the polarization of the wave radiated by the antenna. If the direction is not specied, the direction of maximum gain (see denition later) is understood. Using this denition, the polarization of a dipole antenna is linear and in the same direction as the antenna axis. Illustration: K. El-Darymli Winter /125

30 Radiation Pattern and Related Concepts (ii) Polarization, cont'd It should be pointed out that the earth's presence will aect the antenna pattern, depending on how much the characteristics dier from that of perfect ground. Note that for linearly polarized waves, maximum reception of power will occur when the receiving antenna is aligned along the direction of polarization (e.g., vertical antennas for AM reception; often TV antennas are horizontal; etc.). For circularly polarized waves, orientation of the antenna is not critical as long as its axis is perpendicular to the propagation direction (e.g., most FM is circularly polarized). K. El-Darymli Winter /125 Radiation Pattern and Related Concepts (ii) Polarization, cont'd Circular Polarization Reception: For a circularly polarized wave we want Ẽ 2 = Ẽ 2 x + Ẽ 2 y = a 2 where a is a constant. K. El-Darymli Winter /125

31 Radiation Pattern and Related Concepts (ii) Polarization, cont'd Mathematical Note on Polarization: Suppose we have a plane wave travelling in the + z direction and suppose its Ẽ-eld is given by Ẽ = Ẽ x ˆx + Ẽ y ŷ with Ẽ x = E x0 cos(ωt kz + φ a ) and Ẽ y = E y0 cos(ωt kz + φ b ) where the amplitudes are obvious and the φ's are phase angles. Suppose next that φ b = φ a or φ b = φ a + π. Then Linear Polarization K. El-Darymli Winter /125 Radiation Pattern and Related Concepts (ii) Polarization, cont'd For given case, i.e., when φ b = φ a or φ b = φ a + π, Hence, where ±A is the shape of a line whose points have coordinates ( Ẽ x,ẽ y ) ; i.e., the Ẽ-eld is linearly polarized. What conditions on the phases and amplitudes of Ẽ x and Ẽ y lead to circular polarization of the wave? Prove it! Which is necessary for LHC and which is necessary for RHC polarization? If neither linear nor circular polarization exists, the wave is generally elliptically polarized. K. El-Darymli Winter /125

32 Radiation Pattern and Related Concepts (iii) Principal E- and H-plane Patterns: For linearly polarized antennas, the performance may be described in terms of its principal E- and H-plane patters. The principal E -plane is the plane containing the electric eld vector and the direction of maximum radiation. The principal H-plane is the plane containing the magnetic eld vector and the direction of maximum radiation. The E- and H-eld patterns in these planes are the respective principal plane patterns. Recall, the doughnut-shaped pattern of the current element discussed earlier. K. El-Darymli Winter /125 Radiation Pattern and Related Concepts Class of Radiators: An isotropic radiator An isotropic radiator is a hypothetical lossless antenna having equal radiation in all directions". While no such structure actually exists in nature, it is useful in that the directive properties of realizable antennas may be referenced to this ideal radiator. For example, it is common to nd in the literature the term dbi for db with respect to isotropic to indicate how a particular antenna gain compares to that of an isotropic radiator (more on gain shortly). K. El-Darymli Winter /125

33 Radiation Pattern and Related Concepts Class of Radiators: A directional antenna A directional antenna is one which radiates or receives e-m waves more eciently in some directions than in others. The term is usually applied to antennas whose directivity signicantly exceeds that of a half-wave dipole (more on directivity soon). Diagram (a) on slide?? (60) of this unit shows the pattern of such a directional antenna. The pattern on slide?? (59) is non-directional in any azimuth plane θ = constant (for example, the horizontal plane antenna pattern function, f (φ), θ = π/2, is a circle in the x-y plane) and directional in any elevation plane (with vertical plane pattern given, for example, by g(θ), φ = constant). Such an antenna is termed omnidirectional meaning that it has essentially a non-directional pattern in a given plane (in this case, in azimuth) and a directional pattern in any orthogonal plane (in this case, in elevation) i.e., omnidirectional is a special case of directional. The pattern on slide?? (60), however, is directional, but NOT omnidirectional. K. El-Darymli Winter /125 Radiation Power Density, Radiated Power, and Radiation Resistance The time-averaged Poynting vector in equation (2.20) is, of course, valid for the far-eld of any antenna. That is P a = 1 2 Re { E H } (2.39) where E and H are phasors. The magnitude of this expression is the power density in W/m 2. Thus, to determine the average radiated power, P r, radiated by the antenna, we simply needed to integrate P a over a surface enclosing the antenna. P r = S P a d S (2.40) where, in general, d S is the vector dierential area on the surface (d S = ˆndS where ˆn is the unit normal). K. El-Darymli Winter /125

34 Radiation Power Density, Radiated Power, and Radiation Resistance In view of (2.39), equation (2.40) may be written P r = 1 2 S Re { E H } d S (2.41) For the innitesimal dipole, equation (2.22) gave P r = ηk2 I 0 2 l 2 12π (2.42) K. El-Darymli Winter /125 Radiation Power Density, Radiated Power, and Radiation Resistance or given that k = 2π/λ, P r = η ( π 3 ) I 0 2 ( l λ ) 2. (2.43) However, from our usual view of power we know that the radiated power must also be given by P r = 1 2 I 0 2 R r (2.44) where R r is the so-called radiation resistance i.e., it is the eective resistance which would give rise to power dissipation, P r, when the current magnitude is I 0. K. El-Darymli Winter /125

35 Radiation Power Density, Radiated Power, and Radiation Resistance It is important to realize that R r is NOT ohmic resistance and P r is NOT ohmic power dissipation. Equating the RHS of equations (2.43) and (2.44) give for the innitesimal dipole ( )( ) 2π l 2 R r = η, (2.45) 3 λ and, for free space where η = η 0 120π, R r = 80π 2 ( l λ ) 2. (2.46) K. El-Darymli Winter /125 Radiation Power Density, Radiated Power, and Radiation Resistance We stipulated for this dipole that, as a rule of thumb, l 0.02λ. Thus, even in the best case (i.e., l = 0.02λ ), it is simple to observe that connecting the innitesimal dipole to a practical transmission line having, say, a 50Ω or 75Ω characteristic impedance (Z 0 ) is a very inecient way to radiate energy: Suppose ; then from (2.46),. The gross mismatch with the given Z 0 's and thus the ineciency is obvious. Note that from our transmission line deliberations in Term 6 you should be able to conclude that the reactance of the innitesimal dipole is capacitive. The above example serves to illustrate the fact that short antennas (i.e., short compared to a wavelength or electrically short) have a very low radiation resistance, low eciency and thus as we shall see low gain. An ecient antenna will have a dimension comparable to a wavelength. Therefore, for example, at AM broadcast frequencies of 500 to 1500 khz very high towers are required to carry the antennas. K. El-Darymli Winter /125

36 Radiation Intensity, Directivity and Gain 1. Solid Angle Before discussing the above related quantities, it will be useful to introduce the concept of the solid angle. Recall from two-dimensional geometry that the radian is dened as the measure of the central angle of a circle which subtends an arc length equal to the radius. Trivially, since the total arc length is 2πr there are 2π radians in a complete circle. K. El-Darymli Winter /125 Radiation Intensity, Directivity and Gain 1. Solid Angle, cont'd Extending this idea to three dimensions, we dene a steradian (sr) as the measure of a solid angle (Ω) whose vertex is at the centre of a sphere of radius r and which subtends a surface area of r 2 on the sphere. In general, Ω = S r 2 so when S = r 2, Ω = 1 sr. Since, in spherical coordinates, ds = r 2 sinθdθdφ, dω = ds r 2 = sinθdθdφ. (2.47) Therefore, for the whole sphere, i.e., solid angle for the entire sphere is 4π steradians. K. El-Darymli Winter /125

37 Radiation Intensity, Directivity and Gain 2. Radiation Intensity The radiation intensity, U, is by denition the power radiated per unit solid angle. From equation (2.40), the dierential radiated power is clearly dp r = P a d S over an element of surface d S = dsˆr. Therefore, From (2.47), K. El-Darymli Winter /125 Radiation Intensity, Directivity and Gain 2. Radiation Intensity, cont'd Then, the radiation intensity, by denition, is U = dp r dω = 1 { 2 r 2 Re E } H ˆr (2.48) Remembering that the cross product in (2.32) is in ˆr direction for the far-eld (i.e., P a = P a ˆr), U = r 2 P a From equations (2.47) and (2.48), it may also be seen that (2.49) The radiation intensity of the innitesimal dipole is left as an exercise. K. El-Darymli Winter /125

38 Radiation Intensity, Directivity and Gain 2. Radiation Intensity, cont'd Of course, in general, U in (2.48) and (2.49) may be a function of θ and φ; i.e., U U(θ,φ). However, for an isotropic source, U is a constant, say U 0. The power radiated by such a source becomes, from (2.49) or, equivalently, for an isotropic source, U 0 = P r 4π (2.50) For an non-isotropic source, U 0 is the average radiation intensity (over all directions). K. El-Darymli Winter /125 Radiation Intensity, Directivity and Gain 2. Radiation Intensity, cont'd Finally, we note that a normalized radiation intensity U n (θ,φ) = U(θ,φ) Umax = f (θ,φ) 2 may be used to plot a power pattern f (θ,φ) 2 which has been normalized to unity in the direction of maximum power and Umax is the corresponding maximum radiation intensity. For clarication, it may be noted by way of example that for the innitesimal dipole, f (θ,φ) = sinθ and f (θ,φ) 2 = sin 2 θ gives the normalized power pattern we have now quantied what was discussed in a more descriptive way previously for this dipole. K. El-Darymli Winter /125

39 Radiation Intensity, Directivity and Gain 3. Directivity The directivity, D (or D(θ,φ) since, it is, in general, direction-dependent) is dened as the ratio of the power radiated per unit solid angle to the average power radiated per unit solid angle. From the denition of radiation intensity and average radiation intensity and using equations (2.48) and (2.50), D(θ,φ) = U(θ,φ) U 0 = dp r /dω P r /4π, (2.51) and the directivity is clearly dimensionless. In an exercise, it will be veried that for the innitesimal dipole we get, on using (2.48), (2.50) and (2.51) and our earlier results for P r, that the directivity is given by Note that, ; D(θ,φ) = 1.5sin 2 θ. (2.52) K. El-Darymli Winter /125 Radiation Intensity, Directivity and Gain Directivity for Innitesimal (Hertzian) Dipole (Exercise) See Eqn. (2.21), U = r 2 P a = (A) U 0 = (see eqn. 2.26) Using (A) and (B), (B) as required. K. El-Darymli Winter /125

40 Radiation Intensity, Directivity and Gain 4. Eciency and Gain If the total input power to an antenna is P in and the radiated power is P r, then the radiation eciency, or simply eciency, ε r, is dened as ε r = P r P in = R r R r + R L (2.53) where R L represents the ohmic losses due to conduction eects; and R r radiation resistance. It must be noted that P in is the actual power delivered to the antenna and NOT the power delivered to the transmission line by the power supply. That is, ε r does NOT account for impedance mismatch between the supply line and the antenna, rather it accounts for dielectric eects (ohmic resistance etc.) of the antenna and its surroundings. If an antenna is lossless, ε r = 1. K. El-Darymli Winter /125 Radiation Intensity, Directivity and Gain The gain, G(θ,φ), of an antenna is dened as Therefore, radiation intensity G(θ,φ) = 4π input power = 4π dp r /dω P in G(θ,φ) = ε r dp r /dω P r /4π (2.54) K. El-Darymli Winter /125

41 Radiation Intensity, Directivity and Gain Using (2.51) and (2.54) G(θ,φ) = ε r D(θ,φ) (2.55) The gain is often incorporated into a parameter called the eective isotropic radiated power, EIRP. Note that G 0 is often used for G max, and D 0 for D max The EIRP is dened as EIRP = P in Gmax (2.56) where Gmax is the maximum gain. The importance of equation (2.56) is that, for a given power requirement, P in can be reduced by increasing antenna gain. K. El-Darymli Winter /125 Half-Power Beamwidth The term half-power beamwidth (HPBW) is dened as the angle between the two directions in which the radiation intensity is one-half the maximum as measured in a plane containing the beam maximum. For example, for the small current element discussed earlier, recall that U(θ,φ) sin 2 θ all other parameters being constant. Set If no adjective or descriptor is used with the word beamwidth, it is generally understood to mean the half-power or 3-dB beamwidth. K. El-Darymli Winter /125

42 Reciprocity The question may be protably asked, Does an antenna in reception mode exhibit the same characteristics (i.e., the same antenna pattern parameters) as in transmit mode? The answer lies in the concept of reciprocity. K. El-Darymli Winter /125 Reciprocity Circuit Analogy Consider a two-port network consisting of linear, bilateral (i.e., same looking from both ports), lumped elements. The reciprocity theorem states that if one places a constant voltage (or current) generator across one port and places a current (or voltage) meter across the other port, makes observations of the meter readings and then interchanges the location of the source and the meter, the meter reading will be unchanged (in fact, this is true for any pair of ports in a multiport network). K. El-Darymli Winter /125

43 Reciprocity In a similar fashion, consider the following antenna setups, Case 1 (I B current induced in B, and V A is the corresponding voltage): Case 2 (I A current induced in A, and V B is the corresponding voltage): K. El-Darymli Winter /125 Reciprocity By reciprocity, if V B = V A, then I A = I B. (2.57) Also, dening the transfer impedance in Case 1 as Z AB = V A and in I B Case 2 as Z BA = V B, reciprocity says that Z I AB = Z BA. A Let's prove (2.57) by using the circuit analogy i.e., let's start by representing the antennas and intervening space by a two-port network of linear, passive, and bilateral impedances. Insofar as the input voltage and output current are concerned, such a network may be reduced to an equivalent T-section as shown. Case 1: K. El-Darymli Winter /125

44 Reciprocity Case 2: K. El-Darymli Winter /125 Reciprocity For Case 1, (2.58) Also, (2.59) Substituting (2.58) into (2.59) relates I B to V A and the impedances as I B = V A Z 3 Z 1 Z 2 + Z 2 Z 3 + Z 1 Z 3. (2.60) K. El-Darymli Winter /125

45 Reciprocity Interchanging the source and current meter (Case 2) gives in exactly the same way I A = V B Z 3 Z 1 Z 2 + Z 2 Z 3 + Z 1 Z 3. (2.61) We see from (2.60) and (2.61) that if V A = V B then I A = I B and the reciprocity theorem is veried. REMEMBER, the space between the antennas must NOT exhibit directional properties for this to hold. The idea of reciprocity becomes important in considering mutual impedances due to coupling eects experienced by antennas in close proximity. We shall leave the problems of both self- and mutual-impedance for a later portion of the course. To indicate the magnitude of this problem (in reality), however, it may be pointed out that the mutual impedance development involves the solution of a pair of coupled integral equations. K. El-Darymli Winter /125 Reciprocity More on Reciprocity While magnetic current densities do not exist in nature (i.e. there are no magnetic charges, as far as we know), it is sometimes useful to postulate such entities (this will be discussed in more detail later in the course). If we allow for a magnetic current density M (analogous to current density J), the Maxwell equations may be cast as E = jωµ H M (2.62) H = jωε E + J (2.63) D = ρv (2.64) B = ρm (2.65) where ρ m is the ctitious magnetic charge density. K. El-Darymli Winter /125

46 Reciprocity The Lorentz Reciprocity Theorem: (1) Dierential Form Consider a linear isotropic medium (not necessarily homogeneous) containing two sets of sources (electric and magnetic current densities) J1, M 1 and J 2, M 2. These sources are allowed to radiate, together or singly, giving rise to elds E 1, H 1 and E 2, H 2, respectively. We will now show the validity of the following statement referred to as the Lorentz Reciprocity Theorem (dierential form): [( E1 ) ( H 2 E2 )] H 1 = ( ) ( E1 J 2 + H2 M ) ( ) ( 1 E2 J 1 H1 M ) 2 (2.66) K. El-Darymli Winter /125 Reciprocity The Lorentz Reciprocity Theorem: (1) Dierential Form, cont'd For the rst set of sources and elds E1 = jωµ H 1 M 1 (2.67) H1 = jωε E 1 + J 1 (2.68) and for the second set E2 = jωµ H 2 M 2 (2.69) H2 = jωε E 2 + J 2 (2.70) where we have simply used (2.62) and (2.63) within the given medium for which the permittivity and permeability are ε and µ, respectively. K. El-Darymli Winter /125

47 Reciprocity The Lorentz Reciprocity Theorem: (1) Dierential Form, cont'd Taking the dot product of (2.67) and (2.70) with H 2 and E 1, respectively, gives ( ) H2 E1 = jωµ ( ) E1 H2 = jωε ( H2 H 1 ) ( E1 E 2 ) + ( H2 M ) 1 ( ) E1 J 2 (2.71) (2.72) Subtract (2.71) from (2.72) to get ( ) E1 H2 ( ) H 2 E1 ( ) ( ) ( ) ( = jωε E1 E 2 + jωµ H2 H 1 + E1 J 2 + H2 M ) 1. (2.73) K. El-Darymli Winter /125 Reciprocity The Lorentz Reciprocity Theorem: (1) Dierential Form, cont'd Recalling that for vectors A and B, in general, ( ) B A ( ) A B = ( ) A B, (2.73) may also be written as ( H2 ) E 1 = ( E1 ) H 2 = RHS of (2.73). (2.74) Next, taking the dot product of (2.68) and (2.69) with E 2 and H 1, respectively, and carrying out the above procedure yields ( H1 ) E 2 = ( E2 ) H 1 ( ) ( ) ( ) ( = jωε E2 E 1 + jωµ H1 H 2 + E2 J 1 + H1 M ) 2. (2.75) K. El-Darymli Winter /125

48 Reciprocity The Lorentz Reciprocity Theorem: (1) Dierential Form, cont'd Subtracting (2.75) from (2.74) with an eye on (2.73) gives [( E1 ) ( H 2 E2 )] ( ) ( H 1 = E1 J 2 + H2 M ) ( ) ( 1 E2 J 1 H1 M ) 2 (2.76) as required. This is the dierential form of the Lorentz Reciprocity Theorem. K. El-Darymli Winter /125 Reciprocity The Lorentz Reciprocity Theorem: (2) Integral Form The integral form of (2.76) follows from the divergence theorem. Recall vol ( ) A dv = A d S. (2.77) S From (2.76) we identify [( A = E1 ) ( H 2 E2 )] H 1 K. El-Darymli Winter /125

49 Reciprocity The Lorentz Reciprocity Theorem: (2) Integral Form, cont'd or, equivalently, A = ( E1 J 2 ) + and substituting into (2.77) we may write = vol ( H2 M ) ( ) ( 1 E2 J 1 H1 M ) 2 [( E1 ) ( H 2 E2 )] H 1 d S S [( E1 J 2 ) + ( H2 M ) ( ) ( 1 E2 J 1 H1 M )] 2 dv. (2.78) This is the integral form of the Lorentz Reciprocity Theorem. K. El-Darymli Winter /125 Reciprocity The Diode Probe It is not too hard to argue that on observing the elds in the far eld vol [( ) ( E1 J 2 H1 M )] [( ) ( 2 dv = E2 J 1 H2 M )] 1 dv. (2.79) vol Consider the second source to be a Hertzian dipole (i.e., M 2 = 0) whose current density can be specied by J2 = δ(x x P )δ(y y P )δ(z z P ) p where δ is the Dirac delta function and p is the vector length of the dipole. Then LHS of (2.79) = vol ( E1 J 2 ) dv = E 1 (x P,y P,z P ) p. K. El-Darymli Winter /125

50 Reciprocity The Diode Probe, cont'd Thus, [( ) ( E1 (x P,y P,z P ) p = E2 J 1 H2 M )] 1 dv (2.80) vol where vol contains J 1 and M 1. Now, since E 2 and H 2 at (x 1,y 1,z 1 ) (a point in vol) are the Hertzian dipole elds, and are known, E 1 at the dipole probe position (x P,y P,z P ) can be calculated knowing the sources J 1 and M 1. Exercise:Use (2.80) to show that the far-eld of any nite electric current distribution in free space can have no radial component. K. El-Darymli Winter /125 A Small Current Loop Just as the innitesimal dipole is a good starting point for discussing linear wire antennas, so the small circular current loop will provide an introduction to loop antennas in general. [In this course, we will not consider the general case of larger loops]. This section addresses the far-eld only. Consider the following geometry for a small current loop in the x-y-plane. Remember that the primed coordinates are used for the source points. Here, the loop radius is r 0, R = r r, and d l = r 0 dφ ˆφ. From Chapter 1 of the Term 5 text the transformation from cylindrical to Cartesian coordinates gives for this last quantity = dl = d l = K. El-Darymli Winter /125

51 From equation (1.40), we have the general form of the required vector potential as A( r) = jk r r µ J( r )e v 4π r r dv (2.81) In the case of the small planar loop, we replace J( r )dv with I 0 d l = I 0 r 0 dφ ˆφ so that I 0 d l = I 0 r 0 dφ [ sinφ ˆx + cosφ ŷ ]. (2.82) K. El-Darymli Winter /125 Furthermore, we note: z = 0; since r is in the x-y plane ( i.e., θ = 90 o). K. El-Darymli Winter /125

52 Now, R = r r = the above relationships, R = r r = Therefore, it may be easily seen that [ (x x ) 2 + (y y ) 2 + (z z ) 2 ] 1/2 and, based on R = [r 2 + r 2 0 2rr 0 sinθ(cosφ cosφ + sinφ sinφ )] 1/2. (2.83) In (2.81), we notice a 1 R factor in the amplitude and, since r 0 r for the small loop, we may as well replace R by r for this factor. However, as with the innitesimal dipole, we must be very careful about approximations in the phase term, e jkr. K. El-Darymli Winter /125 We agree to the following: r 2 0 r 2 (even in terms of the phase, this will be OK). However, the third term in (2.83) will have to be considered closely because, while r 2 r 2, the same order does not exist for rr 0 0, r itself being very large. Using the binomial expansion on the remaining part of (2.83), one obtains (after eliminating r 2 0 ) and which implies R r[1 r 0 r sinθ(cosφ cosφ + sinφ sinφ )], kr kr kr 0 sinθ(cosφ cosφ + sinφ sinφ ), e jkr = e jkr e +jkr 0 sinθ(cosφ cosφ +sinφ sinφ ). (2.84) K. El-Darymli Winter /125

53 Also, for the small loop r 0 λ. Therefore, kr 0 = 2πr 0 λ 1. Then, we know from the Maclaurin series that for f (x) 1, e f (x) On this basis, (2.84) becomes e jkr = e jkr [ 1 + jkr 0 sinθ(cosφ cosφ + sinφ sinφ ) ]. (2.85) Noting the several constants (I 0, r 0, etc.) associated with this analysis and using (2.47) and (2.50) along with the approximation in amplitude of R r, (2.81) may be written as A( r) = A( r) = (2.86) K. El-Darymli Winter /125 Since and 2π 2π 0 sinφ dφ = sinφ cosφ dφ = 2π 0 2π cosφ dφ = 0 cosφ sinφ dφ = A( r) = (2.87) Again, recall sin 2 φ = 1 cos2φ 2 and cos 2 φ = 1 + cos2φ 2 A( r) = K. El-Darymli Winter /125

54 We nally have for the vector potential of the small current loop jkµ(i 0 πr 2) A( r) = 0 e jkr sinθ ˆφ (2.88) 4πr In passing, we note that (I 0 πr 2 ) (current times loop area) is the 0 magnitude of a quantity referred to as the magnetic dipole moment, M, of the small current loop and it points perpendicular to the loop in the sense given by the right-hand rule as depicted (i.e., with ngers of the right hand curled in direction of current, M points in the direction of the thumb): Here, then, M = I 0 πr 2 0 ẑ. K. El-Darymli Winter /125 H-Field: Now, with the vector potential established, we may proceed as usual to nd the elds. Using (2.88) and recalling that it is easily seen that Then (DO THIS) H = 1 µ A, H = 1 µr r [ra φ ]ˆθ. H = k 2 (I 0 πr 2 0 )sinθ 4πr e jkr ˆθ (2.89) K. El-Darymli Winter /125

55 E-Field Finally, using the far-eld, source free form of (1.20) it may be easily established (DO IT) that E = ηk 2 (I 0 πr 2 0 )sinθ 4πr e jkr ˆφ (2.90) Poynting Vector, Radiation Intensity, Power and Resistance Without doing the calculations here (BUT THEY SHOULD BE DONE), we have: Time-averaged Poynting Vector: P a = 1 2 Re { E H } =? (2.91) K. El-Darymli Winter /125 Radiation Intensity: U = r 2 P a = η I 0 2 (kr 0 ) 4 sin 2 θ 32 Radiated Power: P r = S P a d S = 2π π 0 0 U sinθdθdφ = πη I 0 (kr 0 ) 4 12 (2.92) Radiation Resistance: R r = 2P ( ) 4 r I 0 = 320π r0 6, (2.93) 2 λ 0 this last expression being given for free-space operation (i.e., when η = η 0 = 120π Ω). K. El-Darymli Winter /125

56 We note that a quick check of the initial power expression or of equation (2.93) itself shows that the radiation resistance can be written in terms of the loop area, A L as ) R r = 31,171( A 2 L λ 4 0 for free-space operation (λ 0 is the free-space wavelength). Finally, if N turns rather than a single loop were used then R r increases by a factor of N 2 so that for free space ( ) 4 r0 R r = 320π 6 N 2. (2.94) λ 0 K. El-Darymli Winter /125 This is because the magnetic dipole moment for N turns increases by a factor of N as compared to a single loop note that the cross product in P a will produce an N 2. The radiation resistance, as usual, is to be distinguished from ohmic loss. If the loop is not lossless, but highly conducting, the ohmic resistance can be determined via the skin depth as discussed in Term 6. Additionally, for several loops, there is an ohmic term due to a phenomenon termed the proximity eect. The latter is documented in graphical form in the literature and may be found in the Balanis reference. As for the innitesimal dipole, the radiation resistance R r for the small loop is very small and such a loop is not generally suited as a transmitting antenna. Small loops may be used as receiving antennas where signal level is large (e.g., multiturn loops are used in portable AM radio reception and single turns are used in pagers) or where noise limitations are such that improving antenna eciency does not necessarily give better reception. K. El-Darymli Winter /125

57 Also, we note that the antenna eciency, ε r, given by equation (2.53) may be written (not just for circular loops) ε r = R r R r + R L (2.95) where R L are losses due to conductiondielectric eects and R r is the radiation resistance. Finally, we note that for this small loop H = H θ ˆθ and E = E φ ˆφ (equations (2.89) and (2.90)), while for the innitesimal dipole E = E θ ˆθ and H = H φ ˆφ (equations (2.16) and (2.17)). However, in both cases, P a will vary as sin 2 θ so that the radiation patterns of the loop have the same general characteristics as the dipole. We note for the loop that the nulls occur at θ = 0, π i.e., along the axis perpendicular to the loop. K. El-Darymli Winter /125 Principal Plane Patterns K. El-Darymli Winter /125