Linear Wire Antennas EE-438/5306 - Antenna Engineering
Outline Introduction Infinitesimal Dipole Small Dipole Finite Length Dipole Half-Wave Dipole Ground Effect Constantine A. Balanis, Antenna Theory: Analysis and Design 4 th Ed., Wiley, 016. Stutzman, Thiele, Antenna Theory and Design 3 rd Ed., Wiley, 01. Linear Wire Antennas
Finite Length Dipole 3
Finite length dipole A finite length dipole is still in the order of a λ, where and a is the thickness of the. However, the length l of the antenna is in the same order of magnitude as the operating wavelength λ < l λ 10 The current distribution is now approximated to a sinusoidal function: I e x, y, z = a z I 0 sin k l z, 0 z l a z I 0 sin k l z, l z 0 Linear Wire Antennas Slide 4
Linear Wire Antennas Slide 5
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Radiated Fields: Element Factor, Space Factor, and Multiplication To obtain the radiated fields of the finite length dipole in the far-field region, we subdivide the antenna into infinitesimal dipoles and integrate to obtain the contributions from all the infinitesimal elements. de θ = jη ki e x, y, z e jkr sin θ dz 4πR R r z cos θ for far-field approximations in phase terms R r for far-field approximations in amplitude terms de θ = jη ki e x, y, z e jkr 4πr sin θ e +jkz cos(θ) dz + l + l ke jkr + l E θ = ඵ deθ = න jη l l 4πr sin θ න Ie x, y, z e +jkz cos θ dz l total field = element factor (space factor) Linear Wire Antennas Slide 7
Radiated Fields Far-Field E θ = jη I 0e jkr πr cos kl cos θ cos kl sin θ H ϕ = E θ η = j I 0e jkr πr cos kl cos θ cos kl sin θ Linear Wire Antennas Slide 8
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Power Density, Radiation Intensity, Radiation Resistance, Directivity W av = 1 Re E H = 1 Re a θe θ a φ E φ η P rad = W av ds = න S 0 π න 0 π a r W av a r r sin θ dθdφ P rad = η I 0 4π C + ln kl C i kl + 1 sin kl S i kl S i kl + 1 cos kl C + ln kl + C i kl C i kl cos y x C i x = න x y dy = න cos y x y dy sin y S i x = න 0 y dy Linear Wire Antennas Slide 10
Power Density, Radiation Intensity, Radiation Resistance, Directivity R r = P rad I 0 = η π C + ln kl C i kl + 1 sin kl S i kl S i kl + 1 cos kl C + ln kl + C i kl C i kl D 0 = F 0ȁ max Q Q = C + ln kl C i kl + 1 sin kl S i kl S i kl + 1 kl cos kl C + ln + C i kl C i kl Linear Wire Antennas Slide 11
Input Resistance To calculate Input Resistance at the terminals, assume lossless antenna (no R L ) and equate the power at the input to the power at the current maximum Assuming a sinusoidal current, I in R in = I 0 R r R in = I 0 I in R r R in = Radiation Resistance at Input terminals R r = Radiation Resistance at Current Maximum I 0 = Current Maximum I in = Current at input terminals R in = sin Introduction to Antennas Slide 1 R r kl
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Half-Wave Dipole 15
Half-Wave Dipole One of the most commonly used antennas. The arms are λ 4 in length and are fed at the center. Radiation Resistance is excellent for transmission line connections: R r = 73 Z in = 73 + j4.5 To get rid of reactance, it is common practice to cut the length until it vanishes. Directivity is also good for omnidirectional terrestrial communications D 0 = 4πU max P rad = 1.643 =.156 dbi Linear Wire Antennas Slide 16
Half-Wave Dipole Fields E θ jη I 0e jkr πr cos π cos θ sin θ H φ j I 0e jkr πr cos π cos θ sin θ W av η I 0 8π r sin3 (θ) U = r W rad η I 0 8π sin3 (θ) Normalized Power Pattern U n sin 3 (θ) Linear Wire Antennas Slide 17
Half-Wave Dipole Introduction to Antennas Slide 18
Half-Wave Dipole Introduction to Antennas Slide 19
Half-Wave Dipole Introduction to Antennas Slide 0
Half-Wave Dipole Introduction to Antennas Slide 1
Dipole - Examples A center-fed dipole of length l is attached to a balanced lossless transmission line whose characteristic impedance is 50 Ω. Assuming the dipole is resonant at the given length, find the input VSWR when (a) l = λ 4 (b) l = λ (c) l = 3λ 4 (d) l = λ VSWR = 1 + ȁγȁ 1 ȁγȁ Γ = R in Z 0 R in + Z 0 R in = sin R r kl R r ቚ λ 6.84 l= 4 R r ቚ λ 73 l= R r ቚ 3λ 37 l= 4 R r ቚ λ = l= 4 Introduction to Antennas Slide
Dipole - Examples The approximate far zone electric field radiated by a very thin wire linear dipole of length l, positioned symmetrically along the z-axis, is given by E θ = C o sin 1.5 (θ) e jkr r Where C o is a constant. Determine the exact directivity and the length of the dipole D 0 = 4πU max P rad π P rad = න 0 න U sin(θ) dθdφ 0 π Introduction to Antennas Slide 3
Dipole Examples A λ dipole with its center at the origin radiates a time-averaged power of 600 W. A second λ dipole is placed with its center point at P(r, θ, φ) where r = 00 m, θ = 90, φ = 40. It is oriented so that its axis is parallel to that of the transmitting antenna. What is the available power at the terminals of the second (receiving) dipole? Assume both antennas are lossless and perfectly matched in all unmentioned parameters. Friis Transmission Equation P r = P t λ 4πr U sin 3 (θ) D 0t D 0r Introduction to Antennas Slide 4