Impedance and Loop Antennas
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1 Impedance and Loop Antennas Ranga Rodrigo University of Moratuwa January 4, 2009 Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
2 Gain Summary of Last Week s Lecture Gain can be defined as maximum radiation intensity G = maximum radiation intensity from a reference antenna with same power output G 0 = maximum radiation intensity from subject antenna radiation intensity from (lossless) isotropic source with same power input (1) (2) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
3 Gain Summary of Last Week s Lecture where k is the efficiency factor. G = kd (3) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
4 Efficiency Summary of Last Week s Lecture Efficiency Efficiency = P out P in = R rad R rad + R loss. (4) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
5 Summary of Last Week s Lecture Effective Aperture Antenna Aperture The ratio of power W in the terminating impedance to the power density of the incident wave will be defined as the effective aperture A e. Thus, where P = P. Thus, A e = Effective aperture = W P = A e, (5) V 2 R T P [ (6) (R rad + R loss + R T ) 2 + (X A + X T ) 2]. V is the induced voltage when the antenna is oriented for maximum response and the incident wave has the same polarization as the antenna. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
6 Summary of Last Week s Lecture Maximum Effective Aperture Maximum Effective Aperture X T = X A (7) R T = R rad. (8) W = V2 R T 4R 2 T = V2 4R rad. (9) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
7 Summary of Last Week s Lecture Maximum Effective Aperture The power W is delivered to the terminating impedance under the conditions of maximum power transfer and zero antenna losses. The ration of this power to the power density of the incident wave is the maximum effective aperture A em. Maximum effective aperture = W P = A em. (10) A em = V2 4PR rad. (11) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
8 Summary of Last Week s Lecture Maximum Effective Aperture of a Hertzian Dipole Maximum Effective Aperture of a Hertzian Dipole A em = 120πE2 (dl) 2 λ 2 320π 2 E 2 (dl) 2 = 3λ2 8π = 0.119λ2. (12) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
9 Summary of Last Week s Lecture Maximum Effective Aperture of a Hertzian Dipole Radiator A em D D (db) Isotropic λ 2 4π Short dipole 1 3λ 2 Half-wave dipole 8π 30λ2 73π A short dipole is always of finite length even though it may be very short. The current along a short dipole is uniform. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
10 Summary of Last Week s Lecture Maximum Effective Aperture of a Hertzian Dipole Received Power in a Communication System P r = P ta et A er. (13) R 2 λ 2 ( ) λ 2 P r = P t G t G r. (14) 4πR Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
11 Self and Mutual Impedances Self and Mutual Impedances: Introduction The impedance presented by an antenna to a transmission line can be represented by a 2-terminal network. The impedance into which the transmission line operates is called the terminal and driving-point impedance. If the antenna is isolated, i.e., remote from ground and other objects, and is lossless, its terminal impedance is is the same as the self-impedance of the antenna. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
12 Self and Mutual Impedances Antenna Transmission line Z Equivalent impedance Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
13 Self and Mutual Impedances Selfimpedance Selfresistance Selfreactance Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
14 Self and Mutual Impedances The self-impedance is the same for reception as for transmission. In the case there are nearby objects, the terminal impedance can still be replaced by a 2-terminal network. However, its value is determined not only by the self-impedance but also by the mutual impedance by it and the other antennas and the currents flowing on them. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
15 Self-Impedance of a Thin Linear Antenna Self-Impedance of a Thin Linear Antenna Assuming that the current distribution at the outset to be nearly sinusoidal for dipoles with length-diameter ratios as small as 100, and that the terminals are at a current maximum, self-impedance is given by Z 11 = R 11 + jx 11 Ω = 30 [ Cin(2πn) + j Si(sπn) ] Ω = 30 [ ln(2πn) Ci(2πn) + j Si(2πn) ] Ω. (15) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
16 Self-Impedance of a Thin Linear Antenna We will state what we mean by Cin, Ci, and Si now. They are three integrals. When we did the radiation resistance calculation, we avoided these integrals by using the numerical result π/2 cos 2 [(π/2) cos θ] sin θ dθ = θ=0 x 1 cos(v) Cin(x) = dv 0 v = ln γx Ci(x) = ln x Ci(x). (16) where γ = e c = or ln γ = c = = Euler s constant. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
17 Ci(x) = sin x. (20) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41 Self-Impedance of a Thin Linear Antenna Cosine Integral The part of this integral given by Ci(x) = ln γx Cin(x). (17) is called the cosine integral. The value of this integral is given by Ci(x) = x cos v v When x is small (x < 0.2), When x is large (x 1), = ln γx x2 2!2 + x4 4!4 x6 6!6 +. (18) Ci(x) ln γx = ln x. (19)
18 Self-Impedance of a Thin Linear Antenna From 16 and 18 we obtain Cin(x) as an infinite series. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
19 Self-Impedance of a Thin Linear Antenna From 16 and 18 we obtain Cin(x) as an infinite series. Cin(x) = x2 2!2 x4 4!4 + x6 6!6. (21) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
20 Self-Impedance of a Thin Linear Antenna Sine Integral The sine integral is given by Si(x) = x 0 sin v v When x is small (x < 0.5), x3 dv = x 3!3 + x5 5!5. (22) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
21 Self-Impedance of a Thin Linear Antenna Sine Integral The sine integral is given by Si(x) = x 0 sin v v When x is small (x < 0.5), x3 dv = x 3!3 + x5 5!5. (22) Si(x) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
22 Self-Impedance of a Thin Linear Antenna Sine Integral The sine integral is given by Si(x) = x 0 sin v v When x is small (x < 0.5), x3 dv = x 3!3 + x5 5!5. (22) Si(x) x. (23) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
23 Self-Impedance of a Thin Linear Antenna Sine Integral The sine integral is given by Si(x) = x 0 sin v v When x is small (x < 0.5), When x is large (x 1), x3 dv = x 3!3 + x5 5!5. (22) Si(x) x. (23) Si(x) π 2 cos x x. (24) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
24 Self-Impedance of a Thin Linear Antenna 2 Ci(x) Si(x) Figure 1: Cosine and Sine Integrals Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
25 Self-Impedance of a Thin Linear Antenna Self-Resistance and Self-Reactance Self-Resistance R 11 = 30 Cin(2πn) Ω = 30 [ ln(2πn) Ci(2πn)] Ω. (25) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
26 Self-Impedance of a Thin Linear Antenna Self-Resistance and Self-Reactance Self-Resistance R 11 = 30 Cin(2πn) Ω = 30 [ ln(2πn) Ci(2πn)] Ω. (25) Self-Reactance X 11 = 30 Si(2πn) Ω. (26) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
27 Self-Impedance of a Thin Linear Antenna These equations give the impedance values for a thin linear center-fed antenna that is an odd number (n) of λ/2. Example Find the self-impedance of a λ/2 antenna. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
28 Self-Impedance of a Thin Linear Antenna These equations give the impedance values for a thin linear center-fed antenna that is an odd number (n) of λ/2. Example Find the self-impedance of a λ/2 antenna. Solution The antenna is a λ/2 antenna. Therefore, n = 1. We have for the self-resistance and self-reactance and R 11 = 30 Cin(2π), (27) X 11 = 30 Si(2π). (28) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
29 Self-Impedance of a Thin Linear Antenna Ci(x) Si(x) Cin(x) Figure 2: Cin Integral Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
30 Self-Impedance of a Thin Linear Antenna Ci(x) Si(x) Cin(x) π Figure 3: Cin Integral Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
31 Self-Impedance of a Thin Linear Antenna R 11 = 30 Cin(2π) = = 73 Ω. (29) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
32 Self-Impedance of a Thin Linear Antenna R 11 = 30 Cin(2π) = = 73 Ω. We can show that, for a λ/2 dipole, Z 11 = R 11 + jx 11 = (29) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
33 Self-Impedance of a Thin Linear Antenna R 11 = 30 Cin(2π) = = 73 Ω. We can show that, for a λ/2 dipole, (29) Z 11 = R 11 + jx 11 = 73 + j42.5 Ω. (30) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
34 Self-Impedance of a Thin Linear Antenna R 11 = 30 Cin(2π) = = 73 Ω. We can show that, for a λ/2 dipole, (29) Z 11 = R 11 + jx 11 = 73 + j42.5 Ω. (30) Resonant Half-Wave Dipole Since X 11 is not zero, an antenna exactly λ/2 long is not resonant. To obtain a resonant antenna, it is common practice to shorten the antenna a few percent to make X 11 = 0. In this case the self-resistance is somewhat less than 73 Ω. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
35 Impedance at a Point Which is Not a Current Maximum Impedance at a Point Which is Not a Current Maximum When we use the methods above, what we obtain is the resistance and reactance at a current maximum. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
36 Impedance at a Point Which is Not a Current Maximum Impedance at a Point Which is Not a Current Maximum When we use the methods above, what we obtain is the resistance and reactance at a current maximum. This is not the point at which the transmission line is connected for an antenna of general length. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
37 Impedance at a Point Which is Not a Current Maximum Neglecting the antenna losses, the value of radiation resistance so obtained is the resistance which would appear at the terminals of a transmission line connected at a current maximum in the antenna, provided that the antenna current distribution on the antenna is the same as when it is center-fed. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
38 Impedance at a Point Which is Not a Current Maximum Neglecting the antenna losses, the value of radiation resistance so obtained is the resistance which would appear at the terminals of a transmission line connected at a current maximum in the antenna, provided that the antenna current distribution on the antenna is the same as when it is center-fed. However, values can be transformed to the value which would appear across terminals of the transmission line connected at the center of the antenna. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
39 Impedance at a Point Which is Not a Current Maximum I1 I 0 A 3λ/4 dipole. x Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
40 Impedance at a Point Which is Not a Current Maximum I1 I 0 A 3λ/4 dipole. x where I = I 0 cos βx. (31) I 1 = terminal current, I 0 = maximum current. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
41 Impedance at a Point Which is Not a Current Maximum I I1 0 x A 3λ/4 dipole. I = I 0 cos βx. (31) where This may be done by equating P rad = I 2 0 R rad/2 to the power supplied by the transmission line I 2 1 R 1/2, where I 1 is the current amplitude at the terminals and R 1 is the radiation resistance at this point. I 1 = terminal current, I 0 = maximum current. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
42 Impedance at a Point Which is Not a Current Maximum where I1 x I 0 A 3λ/4 dipole. I = I 0 cos βx. (31) I 1 = terminal current, I 0 = maximum current. This may be done by equating P rad = I 2 0 R rad/2 to the power supplied by the transmission line I 2 1 R 1/2, where I 1 is the current amplitude at the terminals and R 1 is the radiation resistance at this point. I R 1 = I2 0 2 R rad. (32) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
43 Impedance at a Point Which is Not a Current Maximum We can write Therefore, R 1 = R 1 = ( I0 I 1 ) 2 R 0. (33) R rad cos 2 βx. (34) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
44 Impedance at a Point Which is Not a Current Maximum We can write Therefore, R 1 = R 1 = The reactance X 1 is given by X 1 = ( I0 I 1 ) 2 R 0. (33) R rad cos 2 βx. (34) X rad sin 2 βx. (35) where X rad is the reactance seen at a current maximum point. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
45 Loop Antennas Loop Antennas Loop antennas are simple, inexpensive and very versatile. They may be rectangular, circular, triangular or elliptical. Small loop antennas are defined as those with circumference less thanλ/10 whereas large loops are those with circumference on the order of λ. They are mostly used in HF, VHF and UHF ranges and are commonly used as probes at microwave frequencies. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
46 Loop Antennas Small Loop Antennas Small loop antennas are usually used as receivers rather than transmitters since they are very compact, but not effective in transmitting large amount of power. Portable radios and pagers, where antenna efficiency is not very important as the signal to noise ratio, utilize small loops as receivers. They are also used as probes for field measurement as well as for directional antennas in radio navigation. Reception can be increased by increasing the perimeter, the number of turns, or by inserting a ferrite core (ferrite loop antenna). Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
47 Loop Antennas The Small Loop Antenna The field pattern of a small circular loop of radius a may be determined very simply by considering the square loop of the same area, that is, d 2 = πa 2 (36) where d is the side length of square loop. It is assumed that the loop dimensions are small compare to the wavelenght. The far-field patterns of circular and square loops of the same area are the same when the loops are small but different when they are large in terms of the wavelenght. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
48 Loop Antennas 2a d Figure 4: Circular and Square Loops of Equal Area. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
49 Loop Antennas z y x Figure 5: Orientation of the Square Loop. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
50 Loop Antennas z To distant point θ y d Figure 6: Construction for Finding the Far Field of Dipoles 2 and 4 of Square Loop. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
51 Loop Antennas If the loop is oriented with its plane lying in the xy plane, its far electric field has only an E φ component. To find the far-field pattern in xy plane, it is only necessary to consider two of the four small linear dipoles (2 and 4). Since the individual small dipoles 2 and 4 are nondirectional in the yz plane the field pattern of the loop in this plane is the same as that for two isotropic point sources. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
52 Loop Antennas If the loop is oriented with its plane lying in the xy plane, its far electric field has only an E φ component. To find the far-field pattern in xy plane, it is only necessary to consider two of the four small linear dipoles (2 and 4). Since the individual small dipoles 2 and 4 are nondirectional in the yz plane the field pattern of the loop in this plane is the same as that for two isotropic point sources. E φ = E φ0 e jψ/2 + E φ0 e jψ/2, (37) where E φ0 is the electric field from the individual dipole and ψ = 2πd λ sin θ = d r sin θ, where d r = 2πd λ. (38) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
53 Loop Antennas E φ = 2jE φ0 sin If d λ 38 can be written ( ) dr 2 sin θ. (39) E φ = je φ0 d r sin θ. (40) E φ0 is the electric field due to an individual dipole. These dipoles are short dipoles. We saw that the far-field pattern due to a Hertzian dipole oriented in the z-direction is E = ηβi 0dl sin θ sin(ωt βr)i θ. (41) 4πr where β = 2π λ and η = 120π. A similar formula exists for the short dipole. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
54 Loop Antennas The far-field pattern due a short dipole is E = jηβi 0L sin θ sin(ωt βr)i θ 4πr jηβ[i]l sin θ E θ = 4πr j60π[i] sin θ L = r λ. where L is the length of the dipole, and [I] = I 0 sin(ωt βr), the retarded current. (42) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
55 Loop Antennas The angle θ in the dipole formula is measured from the dipole axis and is 90 in the present case. The angle θ is a different angle with respect to the dipole. Thus we have for the far field E φ0 of the individual dipole E φ0 = j60π[i]l (43) rλ Now, E φ = 60π[I]Ld r sin θ rλ Length of the dipole L = d, d r = 2πd λ, the area of the loop is A = d 2. (44) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
56 Loop Antennas Far Field of a Small Loop Finally, we have E φ = 120π2 [I] sin θ r A λ2. (45) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
57 Loop Antennas Far Field of a Small Loop Finally, we have E φ = 120π2 [I] sin θ r H θ = E φ 120π = π[i] sin θ r A λ2. (45) A λ2. (46) Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
58 Reference Loop Antennas Indra J. Dayawansa. Antennas and propagation. Lecture notes, University of Moratuwa, John D. Kraus. Antennas. McGraw-Hill, John D. Kraus, Ronaled J. Marhefka, and Ahmad S. Khan. Antennas for All Applications. Tata-McGraw-Hill, 3rd edition, Nannapaneni Narayana Rao. Elements of Engineering Electromaganetics. Ranga Rodrigo (University of Moratuwa) Impedance and Loop Antennas January 4, / 41
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