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--- en:electronics:antenna-theory:dipole-derivation [2013/02/19 07:46]
alex
+++ en:electronics:antenna-theory:dipole-derivation [2014/10/20 23:04] (current)
alex
@@ Line 1: / Line 1: @@
 ====== Dipole antenna radiation pattern derivation ======
-For a dipole antenna oriented along the z axis with length L, the far-field radiation pattern can be derived as follows:
+For a dipole antenna centered on the origin and oriented along the z axis with length L, the far-field radiation pattern can be derived as follows:
 The current on the antenna will be approximately sinusoidal, with zeros at the ends of the antenna, represented by
@@ Line 56: / Line 56: @@
 \begin{eqnarray*}
-A_z = \frac{\mu I_0 e^{-jkr}}{4\pi r} \left [ \int_{\frac{-L}{2}}^0 \! \sin \left( k \left( \frac{L}{2} + z' \right) \right) e^{jkz' \cos \theta} \, \mathrm{d}z' + \int_0^{\frac{L}{2}} \! \sin \left( k \left( \frac{L}{2} - z' \right) \right) e^{jkz' \cos \theta} \, \mathrm{d}z' \right ]
+A_z = \frac{\mu I_0 e^{-jkr}}{4\pi r} \left [ \int_{\frac{-L}{2}}^0 \! \sin \left( k \left( \frac{L}{2} + z' \right) \right) e^{jkz' \cos \theta} \, \mathrm{d}z' \right. + \\
+\left. \int_0^{\frac{L}{2}} \! \sin \left( k \left( \frac{L}{2} - z' \right) \right) e^{jkz' \cos \theta} \, \mathrm{d}z' \right ]
 \end{eqnarray*}
@@ Line 124: / Line 125: @@
 \begin{eqnarray*}
-\alpha = \frac{1}{k \sin^2 \theta} \left [ \cos \left( k \left( \frac{L}{2} - z' \right) \right) \cos(kz' \cos \theta) - \sin \left( k \left( \frac{L}{2} - z' \right) \right) \cos(kz' \cos \theta) \cos \theta \right ]
+\alpha = \frac{1}{k \sin^2 \theta} \left [ \cos \left( k \left( \frac{L}{2} - z' \right) \right) \cos(kz' \cos \theta) - \sin \left( k \left( \frac{L}{2} - z' \right) \right) \sin(kz' \cos \theta) \cos \theta \right ]
 \end{eqnarray*}
@@ Line 131: / Line 132: @@
 \begin{eqnarray*}
 A_z = \frac{\mu I_0 e^{-jkr}}{2\pi r k \sin^2 \theta} \left [ \cos \left( k \left( \frac{L}{2} - \frac{L}{2} \right) \right) \cos(k\frac{L}{2} \cos \theta) -\\
-\sin \left( k \left( \frac{L}{2} - \frac{L}{2} \right) \right) \cos(k\frac{L}{2} \cos \theta) \cos \theta - \\
+\sin \left( k \left( \frac{L}{2} - \frac{L}{2} \right) \right) \sin(k\frac{L}{2} \cos \theta) \cos \theta - \\
 \cos \left( k \left( \frac{L}{2} - 0 \right) \right) \cos(k 0 \cos \theta) +\\
-\sin \left( k \left( \frac{L}{2} - 0 \right) \right) \cos(k0 \cos \theta) \cos \theta \right ]
+\sin \left( k \left( \frac{L}{2} - 0 \right) \right) \sin(k 0 \cos \theta) \cos \theta \right ]
 \end{eqnarray*}
@@ Line 139: / Line 140: @@
 \begin{eqnarray*}
-A_z = \frac{\mu I_0 e^{-jkr}}{2\pi r k \sin^2 \theta} \left [ \cos \left ( k\frac{L}{2} \cos \theta \right ) - \cos \left( k \left( \frac{L}{2} \right) \right) \right ]
+A_z = \frac{\mu I_0 e^{-jkr}}{2\pi r k \sin^2 \theta} \left [ \cos \left ( \frac{kL}{2} \cos \theta \right ) - \cos \left( \frac{kL}{2} \right) \right ]
 \end{eqnarray*}
@@ Line 145: / Line 146: @@
 \begin{eqnarray*}
-E_\theta &=& -j \omega \eta A_T \\
+E_\theta &=& -j \omega A_T \\
 A_T &=& A_z \sin \theta \\
-E_\theta &=& \frac{-j \omega \eta \mu I_0 e^{-jkr}}{2\pi r k \sin \theta} \left [ \cos \left (k\frac{L}{2} \cos \theta \right ) - \cos \left( k \left( \frac{L}{2} \right) \right) \right ]
+E_\theta &=& \frac{-j \omega \mu I_0 e^{-jkr}}{2\pi r k \sin \theta} \left [ \cos \left (\frac{kL}{2} \cos \theta \right ) - \cos \left( \frac{kL}{2} \right) \right ]
 \end{eqnarray*}
-This equation for $E_\theta$ is the general form for the theta component in spherical coordinates of the far-field E field of any dipole antenna.
+This equation for $E_\theta$ is the general form for the theta component in spherical coordinates of the far-field E field of a dipole antenna of any length oriented along the z-axis.  The r component is zero due to the far-field assumption and the phi component is zero due to the electric field's orientation along the z-axis.