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Unformatted text preview: 37 Effective area and reciprocity Antenna response to multiple plane waves: • Consider next an arbitrary antenna located at the origin with x y φ R θ z – a resistive termination R , – a gain function G ( θ, φ ) , and Incoherent power addition : Consider a voltage v ( t ) = V 1 cos( ωt ) + V 2 cos( ωt + φ ) applied across a 1 Ω resistor where φ is a random phase shift param eter. Squaring v ( t ) and taking its timeaverage it can be shown that the time averagepower P = 1 T T v 2 ( t ) dt = 1 2 V 2 1 + 1 2 V 2 2 + V 1 V 2 cos( φ ) . Notice the third term in P . With random φ we would be unable to know P , but still its expected value is P = 1 2 V 2 1 + 1 2 V 2 2 , which is the incoherent sum of the timeaverage power due to signals 1 and 2 oneatatime . which is exposed to a spectrum of incident plane wave fields ˜ E i at a frequency f = ω 2 π arriving from directions ( θ i , φ i ) with – power densities S i , – such that each ˜ E i arriving from ( θ i , φ i ) is polarized identically as the field the antenna would radiate toward direction ( θ i , φ i ) . • Assume that each plane wave ˜ E i will deliver an average power P r = S i A ( θ i , φ i ) to the resistor R oneatatime — where A ( θ, φ ) is by definition the effective area of the antenna for reception — and, furthermore, the...
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 Fall '08
 Staff
 Thermodynamics, Polarization, Electromagnet, Resistor, Electrical resistance, Impedance matching

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