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en:electronics:antenna-theory:dipole-derivation [2013/02/19 08:06]
alex
en:electronics:antenna-theory:dipole-derivation [2014/10/20 23:04] (current)
alex
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 ====== Dipole antenna radiation pattern derivation ====== ====== 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 The current on the antenna will be approximately sinusoidal, with zeros at the ends of the antenna, represented by
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 \begin{eqnarray*} \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'​ + \\ +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. ​+ \\ 
-\int_0^{\frac{L}{2}} \! \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*} \end{eqnarray*}
  
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 \begin{eqnarray*} \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*} \end{eqnarray*}
  
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 \begin{eqnarray*} \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) -\\ 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) +\\ \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*} \end{eqnarray*}
  
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 \begin{eqnarray*} \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*} \end{eqnarray*}
  
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 \begin{eqnarray*} \begin{eqnarray*}
-E_\theta &=& -j \omega ​\eta A_T \\+E_\theta &=& -j \omega A_T \\
 A_T &=& A_z \sin \theta \\ 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*} \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. ​ The r component is zero as it is a dipole ​and the phi component is zero due to its orientation along the z-axis.  ​+This equation for $E_\theta$ is the general form for the theta component in spherical coordinates of the far-field E field of 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.  ​