# Antenna Parameters. Ranga Rodrigo. University of Moratuwa. December 15, 2008

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1 Antenna Parameters Ranga Rodrigo University of Moratuwa December 15, 2008 Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

2 Summary of Last Week s Lecture 90 o Radiation Pattern for an Array with 10 Elements 135 o 45 o 180 o φ 0 o φ HP 225 o 315 o 270 o Figure 1: Half-Power Beam Width Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

3 Directivity Summary of Last Week s Lecture Directivity We defined a parameter known as the directivity of the antenna, denoted by the symbol D, as the ratio of the maximum power density radiated by the antenna to the average power density. Thus the directivity of the Hertzian dipole is given by D = [P r] max [P r ] av = 1.5 (1) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

4 Summary of Last Week s Lecture Directivity D = 4π π θ=0 [f (θ, φ)] max 2π f (θ, φ) sin θdθdφ (2) φ=0 Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

5 Summary of Last Week s Lecture Linear Antennas RF generator Z 0 forward wave reverse wave I V max V rms, I rms V min Open circuit transmission line. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

6 Summary of Last Week s Lecture Linear Antennas RF generator Z 0 forward wave reverse wave I V max V rms, I rms V min Radiation from a half-wave dipole. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

7 Summary of Last Week s Lecture Radiation Field Due to half-wave Dipole Now to find the radiation field due to the half-wave dipole, we divide it into a number of Hertzian dipoles, each of length dz. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

8 Summary of Last Week s Lecture Radiation Field Due to half-wave Dipole L 2 z i φ dz z θ θ P r r i θ z cos θ y x L 2 Half-wave dipole as a number of Hertzian dipoles. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

9 Summary of Last Week s Lecture Field Equation for the Half-Wave Dipole Evaluating the integral we obtain E θ = ηi 0 cos [(π/2) cos θ] sin (ωt π ) 2πr sin θ L r. (3a) Similarly where H = H φ i φ. H φ = I 0 cos [(π/2) cos θ] sin 2πr sin θ (ωt π L r ). (3b) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

10 Summary of Last Week s Lecture Power Radiated Power Radiated for the Half-Wave Dipole P rad = 0.609ηI2 0 π sin 2 ( ωt π L r ). (4) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

11 Summary of Last Week s Lecture Power Radiated for the Half-Wave Dipole Time-Average Radiated Power P rad = 1 ( ) 0.609η 2 I2 0. (5) π Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

12 Summary of Last Week s Lecture Power Radiated for the Half-Wave Dipole Radiation Resistance of the Half-Wave Dipole R rad = 0.609η Ω. (6) π For free space, η = η 0 = 120πΩ, and R rad = = 73Ω. (7) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

13 Summary of Last Week s Lecture Radiation Pattern Radiation Pattern of the Half-Wave Dipole Radiation pattern for the field is cos [(π/2) cos θ]. sin θ Radiation pattern for the power desnsity is cos 2 [(π/2) cos θ]. sin 2 θ These are slightly more directional that the corresponding patterns for the Hertzian dipole. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

14 Summary of Last Week s Lecture Radiation Pattern Directivity of the Half-Wave Dipole From 2 D = Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

15 Summary of Last Week s Lecture Center-Fed Linear Antennas of Arbitrary Length Center-Fed Linear Antennas of Arbitrary Length E θ = ηi 0 F(θ) sin(ωt βr). 2πr (8a) H φ = I 0 F(θ) sin(ωt βr). 2πr (8b) R rad = η π π/2 θ=0 F 2 (θ) sin θdθ. [ F 2 (θ) ] max (8c) D = π/2. (8d) θ=0 F2 (θ) sin θdθ Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

16 Summary of Last Week s Lecture Center-Fed Linear Antennas of Arbitrary Length where F(θ) = cos ( βl 2 cos θ) cos βl 2. (9) sin θ F(θ) is the radiation pattern of the fields. For L = kλ, 9 reduces to F(θ) = cos(kπ cos θ) cos(kπ). (10) sin θ Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

17 Antenna Parameters More on Directivity Directivity Continued We learned how to compute the directivity based on the radiation pattern. Directivity has no units, and is usually given in db or dbi, where the reference directivity is 1, the directivity of an isotropic antenna. However, sometimes the reference directivity is that of a half-wave dipole, The directivity then is in dbd. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

18 Gain Antenna Parameters Gain The gain of a transmit antenna is proportional to the directivity by a constant known as the antenna efficiency ot efficiency factor. This efficiency accounts for the power losses within the antenna. Gain, therefore, is not only a descriptor of antenna s power focusing ability, but also, takes into account the power losses within the antenna. Therefore the gain is always less than the directivity. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

19 Gain Antenna Parameters Gain Gain can be defined as maximum radiation intensity G = maximum radiation intensity from a reference antenna with same power output (11) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

20 Gain Antenna Parameters Gain G 0 = Often, the reference antenna is a linear half-wave antenna. If the reference antenna is assumed to be an isotropic antenna with 100 percent efficiency, the gain so defined for the subject antenna is called the gain with respect to an isotropic source and is designated G 0. maximum radiation intensity from subject antenna radiation intensity from (lossless) isotropic source with same power input (12) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

21 Antenna Parameters Let the maximum radiation intensity of from the subject antenna be U m. Let this be related to the value of the maximum radiation intensity U m for a 100 percent efficient subject antenna by a radiation efficiency factor k. Thus Gain U m = ku m (13) where 0 k 1. Therefore 12 may be written as G 0 = U m U 0 = ku m U 0, (14) where U 0 is the radiation intensity from a lossless isotropic source with the same power input. But U m U 0 is the directivity D in terms of intensities. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

22 Gain Antenna Parameters Gain where k is the efficiency factor. G = kd (15) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

23 Antenna Parameters Why Is Gain Always Less Than Directivity For a Practical Antenna? Gain Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

24 Antenna Parameters Why Is Gain Always Less Than Directivity For a Practical Antenna? Gain The gain of a transmit antenna is proportional to directivity by a constant known as the antenna efficiency. This efficiency accounts for the power losses within the antenna. Gain therefore is not only a descriptor of the antenna s power focusing ability but also takes into account power losses within the antenna. Therefore, gain is always less than the directivity. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

25 Efficiency Antenna Parameters Efficiency 1 If the efficiency of an antenna is 100%, all the power delivered to the antenna is radiated. 2 Usually, however, some of this power is lost in conduction and dielectic losses. 3 Therefore, the efficiency is typically less than 100%. 4 A useful antenna should have an efficiency that is in the very least in the 80% range. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

26 Efficiency Antenna Parameters Efficiency P out = I 2 R rad. (16) P in = I 2 (R rad + R loss ). (17) where R loss is the loss resistance. Efficiency = P out P in = R rad R rad + R loss. (18) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

27 Antenna Parameters Antenna Aperture The Aperture Concept Consider a horn antenna as a receiving antenna. The power absorbed by the antenna is SA where A is the aperture area and S is the magnitude of power density of the incident electromagnetic wave. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

28 Antenna Parameters Antenna Aperture Effective Area of an Aperture An antenna has an effective aperture area as a receive antenna; it is the area which extracts the power from the incident electromagnetic wave. As a transmit antenna, the effective area is the area from which electromagnetic waves are given out. For aperture antennas such as horn antennas, the effective area is closely related to the physical area. However, for wire antennas such as dipoles, the effective area is much bigger than the physical area of the antenna. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

29 Antenna Parameters Effective Aperture Antenna Aperture Consider any type of collector or receiving antenna. The antenna collects power from the wave and delivers it to the terminating or load impedance Z T connected to its terminals. The antenna may be replaced by its Thévenin equivalent having an equivalent voltage of V and internal antenna impedance Z A. Z T V Z A Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

30 Antenna Parameters Antenna Aperture The voltage V is induced by the passing wave and produces a current I = V Z T + Z A. (19) In general, the antenna and terminating impedances are complex, thus Z T = R T + jx T (20a) Z A = R A + jx A (20b) The antenna resistance comprises of two parts, a radiation resistance R rad, and a loss resistance R loss, that is R A = R rad + R loss. (21) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

31 Antenna Parameters Antenna Aperture Let the power delivered by the antenna to the terminating impedance be W. Then W = I 2 R T (22) I = V (R rad + R loss + R T ) 2 + (X A + X T ) 2. (23) W = V 2 R T (R rad + R loss + R T ) 2 + (X A + X T ) 2. (24) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

32 Antenna Parameters 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, (25) V 2 R T P [ (26) (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) Antenna Parameters December 15, / 47

33 Antenna Parameters Maximum Effective Aperture Maximum Effective Aperture When the terminating impedance is the complex conjugate of the antenna impedance, so that the maximum power is transferred, and if the antenna losses are zero (R loss = 0 and therefore R A = R rad ), So, the largest possible power is X T = X A (27) R T = R rad. (28) W = V2 R T 4R 2 T = V2 4R rad. (29) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

34 Antenna Parameters 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. (30) A em = V2 4PR rad. (31) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

35 Antenna Parameters Maximum Effective Aperture of a Hertzian Dipole Maximum Effective Aperture of a Hertzian Dipole Consider the Hertzian dipole with length dl λ, with a uniform current distribution. The terminating resistance R T is assumed to be equal to R rad, and the antenna loss resistance R loss is assumed to be equal to zero. V = Edl. (32) We found that ( ) 2 dl R rad = 80π 2 Ω (33) λ Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

36 Antenna Parameters Maximum Effective Aperture of a Hertzian Dipole The (magnitude) of the power density in free space is P = E2 η = E2 120π (34) The maximum effective aperture of the Hertzian dipole is A em = 120πE2 (dl) 2 λ 2 320π 2 E 2 (dl) 2 = 3λ2 8π = 0.119λ2. (35) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

37 Antenna Parameters Maximum Effective Aperture of a Half-Wave Dipole Maximum Effective Aperture of a Half-Wave Dipole L 2 z i φ dz θ r r P z i θ θ z cos θ y x Ranga Rodrigo (University of Moratuwa) L 2 Antenna Parameters December 15, / 47

38 Antenna Parameters Maximum Effective Aperture of a Half-Wave Dipole Maximum Effective Aperture of a Half-Wave Dipole We noticed that when we considered the Hertzian dipole at a distance z from the origin of a half-wave dipole, the current in this Hertzian dipole is I 0 cos(πz /L) cos ωt. For the half-wave dipole L = λ/2. The infinitesimal voltage dv due to the voltage induced by the incident wave is this Hertzian dipole of length dz is dv = E(dz ) cos(2πz /λ) (36) We can find the total induced voltage by integrating over z. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

39 Antenna Parameters Maximum Effective Aperture of a Half-Wave 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) Antenna Parameters December 15, / 47

40 Received Power in a Communication System Received Power in a Communication System The received power in relation to a radio link is given by the Friis transmission formula, which was presented by harold T. Friis while at Bell Labs in Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

41 Received Power in a Communication System Suppose the transmit antenna and the receive antenna are aligned for maximum reception. If the transmit antenna is isotropic, the power density of the electromagnetic wave at the receiver is S r = P t 4πR 2 (37) where P t is the transmit power, and R is the distance from the transmitter. However, if the transmitter has a gain G t, the power density at the receiver is S r = P tg t 4πR 2 (38) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

42 Received Power in a Communication System If the effective area of the receive antenna is A er, the power collected at the receiver, P r = S r A er, is P r = P tg t A er 4πR 2. (39) Directivity and gain are related to the aperture area. Thus, the gain of the transmitter G t is related to its aperture A et by G t = 4π λ 2 A et. (40) Therefore, from Equations 39 and 40 we get P r = P ta et A er R 2 λ 2. (41) Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

43 Received Power in a Communication System Power given by Equation 41 can also be expressed in terms of antenna gains: ( ) λ 2 P r = P t G t G r. (42) 4πR Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

44 Received Power in a Communication System Example 3.1 What is the maximum effective aperture of a microwave antenna with a directivity of 800? Assume that the antenna is designed for 10 GHz. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

45 Received Power in a Communication System Example 3.1 What is the maximum effective aperture of a microwave antenna with a directivity of 800? Assume that the antenna is designed for 10 GHz. Solution 3.2 D = 4π λ 2 A em. λ = 3 cm =.03 m. A em = π = m2. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

46 Received Power in a Communication System Example 3.3 What is the maximum power received at a distance of 2 km over a 7 GHz free-space link consisting of a transmit antenna with a 30 db gain and a receive antenna with a 25 db gain? The gains are expressed with respect to a lossless isotropic source. The transmit antenna input power is 1 kw. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

47 Received Power in a Communication System Solution 3.4 ( ) λ 2 P r = P t G t G r, 4πR P r (dbw) = P t (dbw) + G t (db) + G r (db) + 10 log = , = 12.7 (db), = 0.2 pw. ( λ 4π Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

48 Variation of Directivity with Length Variation of Directivity with Length As the length of the antenna becomes longer, the antenna beamwidth becomes narrower. However, as the antenna becomes longer that a wavelength, side-lobes appear in the radiation pattern. This means that the radiation is now not focused in one main-lobe but is focused in multiple directions. Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

49 Variation of Directivity with Length Reference Indra J. Dayawansa. Antennas and propagation. Lecture notes, University of Moratuwa, John D. Kraus. Antennas. McGraw-Hill, Nannapaneni Narayana Rao. Elements of Engineering Electromaganetics. Prentice Hall, 4th edition, Ranga Rodrigo (University of Moratuwa) Antenna Parameters December 15, / 47

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