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1 MIT OpenCourseWare 6.0/ESD.0J Electromagnetics and Applications, Fall 005 Please use the following citation format: Markus Zahn, 6.0/ESD.0J Electromagnetics and Applications, Fall 005. (Massachusetts Institute of Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). License: Creative Commons Attribution-Noncommercial-Share Alike. Note: Please use the actual date you accessed this material in your citation. For more information about citing these materials or our Terms of Use, visit:

2 6.0 - Electromagnetics and Applications Fall 005 Lecture - Receiving Antennas Prof. Markus Zahn December 6, 005 I. Review of Transmitting Antennas (Short Dipoles) A. Far fields (r λ) z v I (0) dl eff = v I (0) + dl dl I v (z) dz B. Intensity S r Ê 0 Idl eff k η Ê θ = ηĥ φ = sin(θ)e jkr, Ê 0 = ˆ µ, η = jkr 4π ɛ C. Total time average power P S r = [ Re E ˆ ] H ˆ = η Ê π π P = dθ dφ S r r sin(θ) 0 0 Idl eff = ˆ ηk π Ê0 = sin (θ) η kr ˆ 4 η = Idl eff k sin (θ) η 6π k r ˆ k η = Idl eff sin (θ) π r = ˆ R R = πη ( dl eff ) radiation resistance I λ D. Gain S r G(θ, φ) = P /(4πr ) = Idl ˆ eff k η sin (θ) π( πr 4 ) r Idl 8 π ˆ eff η k = sin (θ)

3 II. Receiving Antennas Z S + _ I a d V b c θ E o y In absence of receiving antenna: Ē inc = Ē 0, H inc = H 0. With d λ, over size scale of antenna, Ē 0 and H 0 are approximately spatially uniform. In presence of receiving antenna, electric and magnetic fields are perturbed so that tangential E and normal H are zero along the perfectly conducting length of the antenna. x ϕ e For d << λ E = Ē 0 + Ē () H = H 0 + H () Surface S above intimately hugs the antenna so that ( ) E dā = da Ē 0 + Ē n = 0 (tangential E = 0) () S S +d / da n (H 0 + H ) = da K = dz Ī (z) = I d eff ī z (4) S S d / Another useful relationship: da (Ē 0 H 0) n = ( Ē 0 H 0) da n = 0 (5) S S Integral of normal over closed surface is zero: dv f = da f n, Take f =, f = 0 = da n = 0 (6) Scalar Triple Product Identity: (Interchange of cross and dot) V S S (ā b) c = ā ( b c ) (7) Complex power supplied by receiving antenna P = da Ŝ n (8) S Ŝ = (E ˆ Ĥ ) = [ ] (Ê 0 + Ê ) (Ĥ 0 + Ĥ ) = [ ] Ê 0 (Ĥ 0 + Ĥ ) + Ê (Ĥ 0 + Ĥ ) (9) { [ ] [ ] } P = da Ŝ n = da Ēˆ0 (H ˆ0 + H ˆ ) n + da Ēˆ (H ˆ0 + H ˆ ) n (0) S S S

4 [ ] [ ] da Ê 0 (H ˆ0 + Ĥ ) n = da Ēˆ0 (H ˆ0 + H ˆ ) n S S [ ] = Ēˆ0 da (Ĥ 0 + Ĥ ) n S = Ēˆ0 Î d eff ī z (from (4)) () [ ] [ ] [ ] da Ê (Ĥ 0 + Ĥ ) n = da Ēˆ H ˆ0 n + da Ēˆ H ˆ n () S S S } {{} Î (R+jX) where R is the radiation resistance and X is the antenna reactance [ ] [ ] da Ê H ˆ0 n = da H ˆ0 Ê n S S [ ] = da Ĥ 0 (Ê 0 + Ê ) n S [ ] ( ) = da Ĥ 0 (Ê 0 + Ēˆ) n = H 0 da Ê 0 + Ēˆ n S S = 0 (from ()) () P = [ ] Ê 0 Î d eff ī z + Î (R + jx) = VˆÎ Thevenin Equivalent Circuit VˆT H = Vˆoc = Ēˆ0 d eff (d eff = d eff ī z )

5 III. Transmitting and Receiving Antennas A. Circuit Description I I V V Two port network ˆV = ÎZ + ÎZ ˆV = ÎZ + ÎZ Z = ˆV Î Î =0 Z = ˆV Î Î =0 Reciprocity Theorem: Z = Z Z = R + jx Z = R + jx B. Antenna Thevenin Equivalent I Circuits I V th V Z V * V th Transmitter Receiver (Balanced load Z* to cancel reactance X ) ˆV th = ÎZ = ˆV th = ÎZ = ˆ E dl eff ˆ E dl = ˆ E dl eff 4

6 Vˆth/ = Vˆth = Ê dl eff ( sin( ) θ ) P = R 8 R 8 πη dl eff λ Ê ˆ E sin (θ) λ P = A rec (θ, φ) S r = A rec (θ, φ) = η 8 πη C. Representative Parameters λ λ A rec (θ, φ) = sin (θ) = G rec (θ, φ) 4π 4π. Minimum received power 0 0 watts For total transmitted power of watt, how far away can the receiver be at f = GHz? P trans λ P rec = 4πr G trans G rec 4π }{{}}{{} S r A rec(θ,φ) fλ = c λ = c = 08 =. m f 0 9 G trans = G rec = sin (θ) (for short dipoles) (identical transmitting and receiving antennas) π Take θ = G trans = G rec = ( ) P trans λ r = G trans G rec P rec 4π ( ) ( ) 9. = π = m r = m = km 00, 000 miles. For data transmission, receivers need E b > Joules/bit Power received = ME b where M is the data rate, bits/s 0 9 watts received power allows M = E = b =.5 0 bits/s CD = bytes = bits ( byte = 8 bits) M =.5 0 bits/sec 4.5 CD/sec. Distance is not a barrier to wireless communications r = lightyear = 0 8 m/s 0 7 s/yr = m/yr P trans =? c f = GHz λ = =. m f M = bit/s, E b = Joules/bit P rec = ME b = Watts G trans = G rec = 0 7 5

7 P rec ( ) 4πr λ P trans = Gtrans G rec( ) π(9 0 5 ). = 0 4 = 5 Watts For M =.4 kb/s P trans. MW (with a year delay each way) 4. Optical Communications: E = hf, h = Joule-sec (Planck s Constant) a. Radio Photons b. Optical Photons f = GHz E = Joules/Photon EN = E b N = E b photons/bit E = photons/bit c λ = 0.5 µm f = = Hz E b N = =. photon/bit hf