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--- en:electronics:antenna-theory:dipole-derivation [2013/03/12 02:17] – alex
+++ en:electronics:antenna-theory:dipole-derivation [2014/10/20 21:04] (current) – alex
@@ Line 125: / 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 132: / 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 146: / 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 (\frac{kL}{2} \cos \theta \right ) - \cos \left( \frac{kL}{2} \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 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.