Antenna¶
Antenna Characteristics¶
Radiation Intensity¶
\[
\begin{gather}
U(\theta, \phi) \equiv r^{2} P_{avg}
\end{gather}
\]
-
independent to \(r\)
-
represent the power density at particular direction.
\[
\begin{gather}
P_{rad} = \int_S r^{2}\vec P_{avg}\cdot d\vec S
=
\int_\Omega U \, d\Omega
\\\\
\implies
P_{rad} = 4\pi U_{avg}
\end{gather}
\]
Directive Gain¶
\[
\begin{gather}
G_d(\theta, \phi) = \frac{U(\theta, \phi)}{U_{avg}}
\end{gather}
\]
- normalized version of \(U\).
Directivity¶
\[
\begin{gather}
D = \max\big\{
G_d{(\theta, \phi)}
\big\}
\end{gather}
\]
- sharpness criteria for antenna
Power Gain & Efficiency¶
-
in average
- efficiency
\[ \begin{gather} \eta_r = \frac{P_{rad}}{P_{in}} \end{gather} \]- power gain
\[ \begin{gather} G_{p-avg}=\eta_r = \frac{P_{rad}}{P_{in}} \end{gather} \] -
at particular direction
- power gain
\[ \begin{gather} G_{p}(\theta, \phi) = \frac{4\pi U(\theta, \phi)}{P_{in}} \end{gather} \]
Effective Aperture¶
- isotropy
\[
\begin{gather}
A_{iso} = \frac{\lambda^{2}}{4\pi}
\end{gather}
\]
- effective
\[
\begin{gather}
A_e = A_{iso} D = \frac{\lambda^{2}}{4\pi}D
\end{gather}
\]
Beam Solid Angle¶
\[
\begin{gather}
\Omega_A = \frac{P_{rad}}{U_{max}}
\end{gather}
\]
fff¶
Hertain dipole¶
\[
\begin{align}
H_{\phi s } &= \frac{j\beta I_0 \,dℓ}{4\pi r} \sin \theta \,e^{-j\beta r}
\\\\
E_{\theta s} &= \eta H_{\phi s }
\end{align}
\]
Half wave¶
\[
\begin{align}
H_{\phi s} &= \frac{jI_0 \, e^{-j\beta r}}{2\pi}\frac{\cos\big[\cos\theta\big]}{r\sin\theta}
\\\\
E_{\theta s} &= \eta H_{\phi s }
\end{align}
\]
Radar¶
\[
\begin{align}
𝔓_i &= \frac{P_{rad}}{4\pi r^{2}}G_d
\\\\
𝔓_s &= \lim_{r\to \infty}\frac{𝔓_i\,\sigma}{4\pi r^{2}}
\end{align}
\]