Module - Antenna: Radiation characteristics of antenna, radiation resistance, short dipole antenna, half wave dipole antenna, loop antenna ELL 1 Instructor: Debanjan Bhowmik Department of Electrical Engineering Indian Institute of Technology Delhi Abstract In this module we first introduce a set of parameters that can be used to analyze the quality of an antenna- whether it can radiate power in a particular direction, how much of the incoming energy into the antenna is wasted as heat loss, etc. Next we calculate these parameters for the short dipole antenna, the radiation pattern of which we have analyzed in the previous module. Then we derive the radiation pattern of half wave dipole antenna and loop antenna and calculate these parameters for the same. (Reference: a) Electromagnetics for Engineers- T. Ulaby) b) Antenna Theory- Analysis and Design- by Balanis ) 1
1 Antenna Radiation Characteristics 1.1 Normalized Radiation Intensity In the previous module (Module 1), we have seen that the time averaged power radiated per unit area (power density)= 1 Re( E ( r) H ( r)) (equation 6 of previous module). From equation (61) of Module 1, we can write: Then power per unit area (da)= E ( r) = E ie ikr ˆθ; H ( r) = H ie ikr ˆφ (1) dp rad da = 1 E = 1 Z Z ( ki L r ) sin θ = S(r, θ, φ) = S max F (θ, φ) () for a given radius r, where S max = 1 Z ( kil ) 1 r F (θ, φ) is the normalized radiated power per unit area, also known as normalized radiation intensity. Total time averaged power radiated by antenna through surface of radius r about the source (antenna) is given by: P rad = surface dp π π rad da da = S max F (θ, φ)(r sin(θ)dθdφ) = r S max F (θ, φ)dω (3) θ= φ= Ω Ω F (θ, φ)dω is known as pattern solid angle (Ω P ) 1. Radiation Pattern The normalized radiation intensity F (θ, φ) is a function of both polar angle θ and azimuthal angle φ. The plane of a constant φ is called the elevation plane. In this plane radiation intensity F is a function of different polar angles θ, and can be plotted as a rectangular plot (F on y-coordinate, θ on x-coordinate) (Fig. 1(b)) or polar plot (F as radius r, and θ as angle θ) (Fig. 1(a)). F is plotted in decibels. The difference between two angles θ for which value of F (θ, φ) is half of that of maximum value, which is 1, is known as 3dB beamwidth. From Fig. 1(b) 3dB beamwidth= θ θ 1 The polar plot in Fig. 1(a) shows the presence of radiation lobes. The antenna for which the pattern is plotted here is fairly directive. Most of the energy is radiated in a narrow range, called the main lobe. In addition, pattern exhibits several other lobes called minor lobes. The minor lobes adjacent to the main lobe are called side lobes, while the minor lobes diametrically opposite of the major lobe are called back lobes. Similarly the plane for which polar angle θ is a constant is called the azimuth plane. 1.3 Antenna Directivity By definition of normalized radiation intensity (F ) maximum value of F is 1. The average value of F is given by: F avg = 1 F (θ, φ)dω = 1 Ω P (4)
Figure 1: Radiation pattern of a microwave antenna in (a) polar form (b) rectangular form. Figure from Electromagnetics for Engineers by T. Ulaby 3
Directivity of the antenna is defined as the ratio of maximum normalized radiation intensity (F max ), which is by definition equal to 1, to the average radiation intensity (F avg ). Thus directivity (D) is given as: From equation () above, for a given radius r, S(θ, φ) = S max F (θ, φ) S avg = S max 1 From equation (5) directivity: D = F max F avg = 1 F avg = Ω P (5) Ω F (θ, φ)dω = S max F avg (6) D = 1 F avg = S max S avg (7) So directivity (D) of the antenna can also be defined as the ratio of maximum power density (S max ) to the average power density for a given radius (S avg ). Also, S avg (θ, φ) = 1 S max F (θ, φ)dω = 1 r S max r F (θ, φ)dω = P rad r (8) from equation (3). Thus directivity can also be written as: Ω Ω D = S max S avg = r S max P rad (9) 1.4 Radiation Efficiency, Antenna Gain and Radiation Resistance Ratio of power radiated by the antenna (P rad ) to the total power transmitted to the antenna (P total ) is known as the radiation efficiency (ξ). ξ = P rad P total (1) Multiplying radiation efficiency with directivity we get the gain of the antenna (G) as follows: G = ξd = r S max P total (11) Radiation resistance (R rad ) is defined such that time averaged radiated power can be written as: P rad = 1 (I )R rad (1) Similarly, the loss resistance (R loss ) is defined time average power dissipated in antenna as heat loss can be written as: P loss = 1 (I )R loss (13) Efficiency ξ can be written as: ξ = P rad P total = P rad P rad + P loss = 4 R rad R rad + R loss (14)
1.5 Evaluation of antenna parameters for short dipole antenna We now calculate the parameters we defined in this section for a short dipole antenna, the radiation pattern of which we already derived in previous Module. For short dipole antenna, the time averaged power radiated per unit area is given by: Sp, S(r, θ, φ) = 1 Z ( ki L r ) sin θ = S max F (θ, φ) (15) S max = 1 Z ( ki L r ) ; F (θ, φ) = sin θ (16) Pattern solid angle π π Ω P = F (θ, φ)dω = sin (θ)(sin(θ)dθdφ) = 8π (17) φ= 3 Directivity θ= D = Ω P = 1.5 (18) F (θ, φ) is half of maximum value (1) when θ=45 degree and - 45 degree. So 3dB beamwidth = 45- (-45)= 9 degrees. Time-average radiated power P rad = r S max Ω F (θ, φ)dω = r S max 8π 3 = 1 Z ( ki L ) 8π 3 = 1 Z ( I L λ ) 8π 3 Comparing with expression for radiation resistance R rad (equation 1) = Z ( I L λ ) π 3 (19) R rad = π 3 (L λ ) Z () For a given R loss, radiation efficiency ξ and gain (G) are given as: Half wave dipole ξ = R rad R rad + R loss ; G = Dξ (1) In a half wave dipole antenna, the alternating signal is applied at the center of the dipole. At the two ends of the dipole the current is zero from boundary conditions.there is a standing wave formed for both current and voltage inside the dipole. Here we do not derive how current and voltage vary along the dipole. We will do it in the next module, when we discuss transmission line in the next module. Let z direction be the direction of length of the dipole. For a half wave dipole λ 4 z λ 4 () 5
i(z, t) = I cos(ωt)cos(kz) = Re(I(z)e iωt (3) For each small current element of length dl with current I(z), we already know, from the previous module (derivation of electric and magnetic fields radiated from a short dipole antenna) that at a point away from the antenna with coordinates (r, θ, φ) (coordinates with respect to center of antenna) or (s, θ s, φ s ) (coordinates with respect to that small current element) de = ic µ ki(z)dz e iks sin(θ s ) s ˆθ s (4) We can take the following approximations: 1 s 1 r, θ s θ, ˆθ s ˆθ (5) But we cannot assume e iks e ikr because of the complex exponential factor. This leads to a phase error. Instead assuming, we get The net electric field for the entire dipole: E = λ 4 λ 4 where Z = s = (r zcos(θ)) + (zsin(θ)) r zcos(θ) (6) e iks e ikr e ikzcos(θ) (7) ic µ k r I cos(kz)e ikr e ikzcos(θ) sin(θ)dz ˆθ = ic µ λ k r I e ikr 4 sin(θ) cos(kz)e ikzcos(θ) ˆθdz λ 4 µ ɛ = ic µ k r I e ikr sin(θ)( k cos( π cos(θ)) sin )ˆθ (θ) 1 I cos(θ))e ikr = iz π sin(θ) cos(π ˆθ (8) r is the characteristic impedance of free space.some trignonometry steps are skipped. Hence the magnetic field for the entire dipole is: Power per unit area is given by H = i 1 I cos(θ))e ikr π sin(θ) cos(π ˆφ (9) r S(r, θ, φ) = I Z π 8π r (cos( (cos(θ)) ) (3) sin(θ) Comparing the expression for power radiated for half wave dipole we obtained here with power radiated for short dipole, we make the following observations: 1. Power radiated and radiation resistance are not dependent on the ratio of length of dipole to wavelength. Since for short dipole, length to wavelength ratio is very small power radiated is also 6
very small. However length to wavelength ratio is 1/, as a result the final expression for power and radiation resistance do not have length dependence. But power radiated and radiation resistance of half wave dipole are much larger than short dipole.. The normalized radiation density function F (θ, φ) is more complex here than in the case of short dipole. But the polar plots of the two are almost identical (check the homework problem where you have done this numerically). 3. The directivity of half wave dipole can be evaluated numerically usingf (θ, φ). The directivity comes out to be 1.64, slightly higher than short dipole. 3 Dipole of finite length For an antenna of arbitrary length L, the following boundary conditions are satisfied: I( L ) =, I( L ) = (31) Standing wave is formed like before, the current distribution along the line is given by: I(z) = I sin(k( L z)) z L ; I(z) = I sin(k( L + z)) L z (3) Total electric field phasor: E = ic µ k r I e ikr sin(θ)[ sin(k( L L + z))eikzcos(θ) dz + L sin(k( L z))eikzcos(θ) dz] (33) L = 1 sin(k( L z))eikzcos(θ) dz = L L sin(k( L z))(cos(kzcos(θ)) + isin(kzcos(θ)))dz [(sin(k( L z)+kzcos(θ))+sin(k(l z) kzcos(θ)))+i(cos(k(l z) kzcos(θ)) cos(k(l z)+kzcos(θ)))]dz = 1 L ([ cos(k( z) + kzcos(θ)) ] L kcos(θ) k + [ cos(k( L z) kzcos(θ)) kcos(θ) k + i[ sin(k( L z) kzcos(θ)) ] L kcos(θ) k i[ sin(k( L z) + kzcos(θ)) ] L kcos(θ) k ) = 1 kl (cos( ) cos( kl cos(θ)) + cos( kl cos(θ)) cos( kl ) k cos(θ) 1 cos(θ) + 1 + i sin( kl ) + sin( kl cos(θ)) cos(θ) + 1 + i sin( kl cos(θ)) sin( kl ) ) (34) 1 cos(θ) ] L 7
L sin(k( L + z))eikzcos(θ) dz = L = 1 L ([ cos(k( + z) + kzcos(θ)) kcos(θ) + k + i[ sin(k( L + z) kzcos(θ)) k kcos(θ) sin(k( L + z))(cos(kzcos(θ)) + isin(kzcos(θ)))dz ] L ] L + [ cos(k( L + z) kzcos(θ)) k kcos(θ) ] L i[ sin(k( L + z) + kzcos(θ)) ] ) k + kcos(θ) L = 1 kl (cos( cos(θ)) cos( kl ) + cos( kl cos( kl cos(θ)) k cos(θ) + 1 cos(θ) 1 + i sin( kl ) sin( kl cos(θ)) 1 cos(θ) i sin( kl ) + sin( kl cos(θ)) ) (35) 1 + cos(θ) Then electric field phasor: E = ic µ kl k r I ikr (cos( sin(θ)e ) cos( kl cos(θ))) 1 ( k cos(θ) 1 1 cos(θ) + 1 )ˆθ Similarly the magnetic field phasor is given by: = i Z kl I cos( cos(θ)) (cos( kl ) e ikr πr sin(θ) ˆθ (36) H = i I kl cos( cos(θ)) (cos( kl ) e ikr ˆφ (37) πr sin(θ) (Magnetic fields due to each current element of length dl add up to give the net magnetic field the same way as electric field) Power density/ power radiated per unit area is given by: S(r, θ, φ) = I kl Z 8π (cos( cos(θ)) cos( kl ) ) (38) r sin(θ) This function can be evaluated numerically for antennas of different lengths L and then polar plot of normalized radiation intensity (F (θ, φ)), directivity, 3 db beamwidth etc. can be obtained. Please refer to the Radiation patterns of long dipole antennas and large loop antenna lecture slides for that. 4 Small loop antenna Let us consider a circular loop of radius a placed on the x-y plane with the center being at the origin of the Cartesian coordinate system (x,y,z). Any point on the loop/ring is given by (x,y,z ) where x = acos(φ ), y = asin(φ ), z =. For a given angle φ, the small section of the ring at that position carries current as given below: di = Re(I e iωt ˆφ = I e iωt ( sin(φ )ˆx + (cos(φ )ŷ)) (39) 8
Current is assumed constant throughout the loop. Any point (x,y,z) where we want to find the potential and the fields is given by (x,y,z) or (r, θ, φ) where x = rsin(θ)cos(φ), y = rsin(θ)sin(φ), z = rcos(θ) We can write: ˆx = sin(θ)cos(φ)ˆr + cos(θ)cos(φ)ˆθ sin(φ) ˆφ; ŷ = sin(θ)sin(φ)ˆr + cos(θ)sin(φ)ˆθ + cos(φ) ˆφ (4) Now expressing the current element in (r, θ, φ) coordinates d I = Re(I e iωt ( sin(φ )ˆx + cos(φ )ŷ)) = Re(I e iωt ( sin(φ )(sin(θ)cos(φ)ˆr+cos(θ)cos(φ)ˆθ sin(φ) ˆφ)+cos(φ )(sin(θ)sin(φ)ˆr+cos(θ)sin(φ)ˆθ+cos(φ) ˆφ)) = Re(I e iωt (sin(θ)sin(φ φ )ˆr + cos(θ)sin(φ φ )ˆθ + cos(φ φ )φ) (41) Distance between point on the ring/ loop (x y,) and point (x,y,z) is given by : s = ((x x ) ) + ((y y ) ) + ((z z ) ) = (r ) + (a ) (rasin(θ)cos(φ φ )) (4) Hence vector potential due to all such current elements (dl ) in the loop/ ring is: A = Re( µ = Re( µ I e iks s dl φ =π φ = I e iωt (sin(θ)sin(φ φ )ˆr + cos(θ)sin(φ φ )ˆθ + cos(φ φ )φ)) e ik r +a rasin(θ)cos(φ φ ) r + a rasin(θ)cos(φ φ ) adφ (43) Let f(a) = e ik r +a rasin(θ)cos(φ φ ) r + a rasin(θ)cos(φ φ ) (44) Thus, f() = e ikr ; f () = df = ( ik r da a= r + 1 r )sin(θ)cos(φ )e ikr (45) f(a) = f() + f ()a = ( 1 r + a(ik r + 1 r )sin(θ)cos(φ ))e ikr (46) The potential and fields should be independent of φ coordinate at the point where we want to calculate potential and fields owing to symmetry of the problem. The ring/ loop is a circle of radius a on x-y plane about the origin. Hence, A φ = Re( µ ai π e i(ωt kr) ( 1 r + a(ik r + 1 r )sin(θ)cos(φ ))cos(φ )dφ = Re( µ ai e i(ωt kr) a( ik 4 r + 1 r )sin(θ)); A θ =, A r = (47) 9
Only considering the 1 r B = Re(ˆθ( µ a I 4r B = A 1 = ˆr rsin(θ) ( θ (A φsin(θ)) + ˆθ(( 1 r ) r (ra φ)) (48) component of the field, sin(θ)e i(ωt) r (e ikr (ik + 1 r )) = µ k a I sin(θ) cos(ωt kr)ˆθ (49) 4r E = Re( c iω ( B)) = Re( c 1 iω r r (rb θ)) = cµ k a I sin(θ) cos(ωt kr) 4r ˆφ (5) The expressions for electric field and magnetic field for loop antenna are similar to that of short dipole antenna, but for loop antenna electric field is along ˆφ and magnetic field is along ˆθ. Power density/ power radiated per unit area: S(r, θ, φ) = 1 Re( E H ) = cµ ((ka) 4 )I (sin(θ) ) 3r = Z ((ka) 4 )I (sin(θ) ) 3r (51) The normalized radiation intensity function is identical to that of short dipole antenna. Directivity=1.5 Total power radiated= P rad = Z π π 3 ((ka)4 )(I ) φ= θ= F (θ, φ) = sin (θ) (5) sin (θ) r r sin(θ)dθdφ = Z π 1 ((πa λ )4 )I (53) As opposed to ( l λ ) factor for short dipole power radiated by loop antenna depends on ( a λ )4, making the power very small. If there are N turns in the loop, then electric field gets multiplied by N times and magnetic field gets multiplied by N times making the power go up by N times. For large loop antennas, the current distribution cannot be assumed constant throughout the loop. Nodes and anti-nodes are formed. The field pattern for large loop antennas can be solved numerically. Results can be found in the Radiation patterns of long dipole antennas and large loop antenna lecture slides. 1