Find a criterion for the argument that a receiving antenna actually is in the far field of a transmitting antenna. Estimate the distance d of the far zone if D = 10 cm and λ = 3 cm. Assume that they have antenna gains GA = 1.5 and GB = 1.64, respectively, and find the ratio Pr/Pt, at that distance.

Question

Find a criterion for the argument that a receiving antenna actually is in the far field of a transmitting antenna. Estimate the distance d of the far zone if D = 10 cm and λ = 3 cm. Assume that they have antenna gains [latex]G_{A }= 1.5[/latex] and [latex]G_{B} = 1.64[/latex], respectively, and find the ratio [latex]P_{r} / P_{t}[/latex], at that distance.

Accepted Answer

For the Friis transmission equation to be applicable, both antennas must be in the far field. The receiving antenna will be in the far field if the incident spherical wave deviates from an actual plane wave by less than a small fraction of a wavelength. Assume the largest dimension of the receiving antenna is D. By convention, we assume this deviation is approximately Δ ≈ λ/16, which means a phase difference of 22.5°. From the figure, we write

[latex]R^{2} = (R - Δ)^{2} + \left\lgroup\frac{D}{2} \right\rgroup ^{2} \approx R^{2} - 2R\Delta + \frac{D^{2} }{4}[/latex]

This implies that the receiving antenna will be in the far field, or the Fraunhofer zone, if the second term is at least comparable with the third one or [latex]R ≈ D^{2} /8Δ[/latex]. This means

[latex]R ≈ \frac{2D^{2}}{λ}[/latex]

For the particular values, we obtain

[latex]R = \frac{2× 0.1^{2}}{0.03 } = 66.7 cm[/latex]

Beyond this distance, the receiving antenna will be in the far-field region of the transmitting antenna.
The Friis equation (8.55) gives the ratio

[latex]\frac{P_{r}}{P_{i}} = \frac{A_{e,t} A_{e,r}}{\lambda ^{2} R^{2}} (8.55)[/latex]

[latex]\frac{P_{r}}{P_{i}} = \frac{1.07 × 1.17}{16.7^{2} × 3^{2}} = 5 × 10^{-4} [/latex]

The degradation of a received signal is approximately

[latex]-10 \log_{10} ( 5 × 10^{-4}) = 33 dB[/latex]

Question 8.13: Find a criterion for the argument that a receiving antenna a...

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