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Antenna¶

Antenna Characteristics¶

Radiation Intensity¶

\begin{matrix} U (θ, ϕ) \equiv r^{2} P_{a v g} \end{matrix}

independent to $r$
represent the power density at particular direction.

\begin{matrix} P_{r a d} = \int_{S} r^{2} {\vec{P}}_{a v g} \cdot d \vec{S} = \int_{Ω} U d Ω \\ ⟹ P_{r a d} = 4 π U_{a v g} \end{matrix}

Directive Gain¶

\begin{matrix} G_{d} (θ, ϕ) = \frac{U (θ, ϕ)}{U_{a v g}} \end{matrix}

normalized version of $U$ .

Directivity¶

\begin{matrix} D = max {G_{d} (θ, ϕ)} \end{matrix}

sharpness criteria for antenna

Power Gain & Efficiency¶

in average
- efficiency
$\begin{matrix} η_{r} = \frac{P_{r a d}}{P_{i n}} \end{matrix}$
- power gain
$\begin{matrix} G_{p - a v g} = η_{r} = \frac{P_{r a d}}{P_{i n}} \end{matrix}$
at particular direction
- power gain
$\begin{matrix} G_{p} (θ, ϕ) = \frac{4 π U (θ, ϕ)}{P_{i n}} \end{matrix}$

Effective Aperture¶

isotropy

\begin{matrix} A_{i s o} = \frac{λ^{2}}{4 π} \end{matrix}

effective

\begin{matrix} A_{e} = A_{i s o} D = \frac{λ^{2}}{4 π} D \end{matrix}

Beam Solid Angle¶

\begin{matrix} Ω_{A} = \frac{P_{r a d}}{U_{m a x}} \end{matrix}

fff¶

Hertain dipole¶

\begin{aligned} H_{ϕ s} & = \frac{j β I_{0} d ℓ}{4 π r} \sin θ e^{- j β r} \\ E_{θ s} & = η H_{ϕ s} \end{aligned}

Half wave¶

\begin{aligned} H_{ϕ s} & = \frac{j I_{0} e^{- j β r}}{2 π} \frac{\cos [\cos θ]}{r \sin θ} \\ E_{θ s} & = η H_{ϕ s} \end{aligned}

Radar¶

\begin{aligned} 𝔓_{i} & = \frac{P_{r a d}}{4 π r^{2}} G_{d} \\ 𝔓_{s} & = lim_{r \to \infty} \frac{𝔓_{i} σ}{4 π r^{2}} \end{aligned}