Notes5 - Notes#5 ECE594I Fall 2009 E.R Brown Free-Space...

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Notes #5, ECE594I, Fall 2009, E.R. Brown 93 Free-Space Power Coupling for Two Special Cases: Radar and Radiometry Friis' Transmission Formulation Marconi was the pioneer for a new generation of electrical engineers working in the area of “wireless”. One of the truly brilliant amongst these was Friis working at Bell Laboratories in the 1920s and 30s. Among other things, Friis was the first to take advantage of the inherent nature of antennas as passive, reciprocal components, and treat the free-space propagation between a transmit antenna and receive antenna as a two-port “link”. This was done first and foremost for the wireless communications “link”, which we review here first to set the stage for the following RF and THz sensor (i.e., radar and radiometer) “link” formulation. The first step in Friis’ formulation is the concept of an effective aperture A eff for the receiving antenna, p r r inc eff rec S A P ε φ θ = ) , ( (1) where P rec is the power available to the antenna for delivery to a load, ) , ( r r inc S is the average Poynting vector for incoming radiation along the direction ( θ r r ) in the spherical coordinates centered at the receiving antenna, and ε p is the polarization coupling efficiency. Note that this expression applies only when ) , ( r r inc S is aligned with the direction of the beam-pattern maximum . When there is misalignment, another factor is required which is the just the receive beam-pattern, p r r inc r r r eff rec S F A P = ) , ( ) , ( . (2) Next, we suppose that this received Poynting vector is generated by a second, transmitting antenna. We can relate the received power to the properties of the transmitting antenna by: ) ( 4 ) , ( ) ( 4 ) , ( ) , , ( ) , ( 2 2 r r F G P r r F D P r S S t t t t in t t t t rad t t t r r inc τ π = (3)
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Notes #5, ECE594I, Fall 2009, E.R. Brown 94 where the subscript "t" is for transmitting, P rad is the total radiated power, P in is the power used to drive the transmitting antenna (in the matched case, equal to P rad ), θ t and φ t are the spherical angles in the spherical coordinate system centered at the transmitting antenna, τ (r) is the power transmission function including all attenuation effects, and r is the distance (i.e., the “range” between transmitter and receiver. In writing (3) it is understood that F t is taken in the direction ( θ t , φ t ) pointing towards the receiver, which is not necessarily the direction of the maximum of F t . Substitution of (3) into (2) yields the relationship p r r t t t r t in eff rec r r F F G P A P ε τ π φ θ = ) ( 4 ) , ( ) , ( 2 (4) This can be simplified further in terms of the (ostensibly) known parameters of the receiving antenna using the relationships, 2 eff r rec r r rec out / A 4 G P D G P P λ π = (5) where P out is the power delivered to the load of the receiving antenna. Substitution of (4) into (5) yields p t t t r r r t r in out r F F G G r P P λ = ) ( ) , ( ) , ( 4 2 1 (6) the expression commonly known as Friis' formula after its originator. It effectively treats the
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Notes5 - Notes#5 ECE594I Fall 2009 E.R Brown Free-Space...

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