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en:electronics:antenna-theory:dipole-derivation [2013/02/19 07:25] 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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| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Find the vector potential in the Z direction from the current distribution | + | Use the current distribution to find the Z component of the vector potential |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| Line 41: | Line 41: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | As $ r >> z' \cos \theta $, $ z' \cos \theta $ can be neglected in the denominator. However, it cannot be ngeglected in the exponential as it is a phase offset. | + | As $ r >> z' \cos \theta $, $ z' \cos \theta $ can be neglected in the denominator. However, it cannot be neglected in the exponential as it is a phase offset. |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| Line 53: | Line 53: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Elimiate the absolute value by splitting into two integrals: | + | Eliminate the absolute value by splitting into two integrals: |
| \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' + \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*} | \end{eqnarray*} | ||
| Line 65: | Line 66: | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
| - | Substitute $ \frac{e^{jx} + e^{-jx}}{2} = cos(x) $ | + | Substitute $ e^{jx} + e^{-jx} = 2 \cos(x) $ |
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| Line 102: | Line 103: | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| \alpha = \overbrace{\sin \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\sin(kz' \cos \theta)}{k \cos \theta} - \cos \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\cos(kz' \cos \theta)}{k \cos^2 \theta}}^{\beta} + \\ | \alpha = \overbrace{\sin \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\sin(kz' \cos \theta)}{k \cos \theta} - \cos \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\cos(kz' \cos \theta)}{k \cos^2 \theta}}^{\beta} + \\ | ||
| - | \underbrace{\int \sin \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\cos(kz' \cos \theta)}{\cos^2 \theta} \mathrm{d}z'}_{\frac{\alpha}{\cos^2 \theta}} | + | \underbrace{\int \sin \left( k \left( \frac{L}{2} - z' \right) \right) \frac{\cos(kz' \cos \theta)}{\cos^2 \theta} \mathrm{d}z'}_{\alpha / \cos^2 \theta} |
| \end{eqnarray*} | \end{eqnarray*} | ||
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| \alpha &=& \beta + \frac{\alpha}{\cos^2 \theta} \\ | \alpha &=& \beta + \frac{\alpha}{\cos^2 \theta} \\ | ||
| \alpha \cos^2 \theta &=& \beta \cos^2 \theta + \alpha \\ | \alpha \cos^2 \theta &=& \beta \cos^2 \theta + \alpha \\ | ||
| - | \alpha &=& \frac{\beta \cos^2 \theta}{\cos^2 \theta -1} | + | \alpha &=& \beta \frac{\cos^2 \theta}{\cos^2 \theta -1} |
| \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 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. | ||