Sample, Final exam #2

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4. What is the total power radiated by an isotropic source in free space if |E|=1 mV/m at a distance of 10 km?
5. A dipole antenna is 1m long and is fed with a current of magnitude 2A. a. Find the radiated power of the antenna at 540 kHz (lowest AM frequency). b. Find the radiated power if operated at 1200 MHz (low microwave range). Solution: The radiated power may only be calculated if we know the type of antenna we have. For this we must first evaluate the wavelength and after that find the antenna radiation resistance. a. At 540 kHz: λ = c f = 3×10 8 540×10 3 = 555.6 m The antenna is clearly a Hertzian dipole and we may use the radiation resistance in Eq. (18.59). With this we can write: b. At 88 MHz: P rad = I 0 2 R rad 2 = I 0 2 2 80π 2 ⎜ ⎛ ⎝ ∆l' λ ⎟ ⎞ ⎠ 2 = 2 2 2 ×80×π 2 × ( ) 1 555.6 λ = c f = 3×108 88×10 6 = 3.41 m 2 = 5.11 mw This antenna is not a Hertzian dipole because it is not short enough. It is not particularly long either so we will use the expression for the Hertzian dipole: c. At 180 MHz: P rad = I 0 2 R rad 2 = I 0 2 2 80π 2 ⎜ ⎛ ⎝ ∆l' λ ⎟ ⎞ ⎠ λ = c f = 3×10 8 2 = 2 2 2 ×80×π 2 × ( ) 1 3.41 180×10 6 = 1.667 m 2 = 135.8 w This antenna is not a Hertzian dipole. In fact, forL=1m=0.6λ, β=2π/λ: and we must use the expression for arbitrarily long antennas: P rad = ηI 0 2 θ=π 4π∫ ( ( ) ) θ=0 2 cos (βL/2)cosθ − cos(βL/2) dθ sinθ w From Eq. (18.139) and Table 18.2, the value of the integral above is: ⌠ ⌡ θ=π θ=0 ( ( ) ) cos (βL/2)cosθ − cos(βL/2) 2 sinθ dθ = 1.8535
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Sample, Final exam #2

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