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Analytical comparisons
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This section presents comparisons between analytical solutions and modelled solutions using gprMax.

Hertzian dipole in free space
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:download:`hertzian_dipole_fs.in <../../tests/analytical/hertzian_dipole_fs/hertzian_dipole_fs.in>`

This example is of a Hertzian dipole, i.e. an additive source (electric current density), in free space.

.. literalinclude:: ../../tests/analytical/hertzian_dipole_fs/hertzian_dipole_fs.in
    :language: none
    :linenos:

The function ``hertzian_dipole_fs``, which can be found in the ``analytical_solutions`` module in the ``tests`` sub-package, computes the analytical solution.

Results
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:numref:`hertzian_dipole_fs_results` shows the time history of the electric and magnetic field components of the modelled and analytical solutions. The responses completely overlap one another due to their similarity. Therefore, :numref:`hertzian_dipole_fs_results_diffs` shows the percentage differences between the modelled and analytical solutions.

.. _hertzian_dipole_fs_results:

.. figure:: ../../tests/analytical/hertzian_dipole_fs/hertzian_dipole_fs_vs_analytical.png
    :width: 600 px

    Time history of the electric and magnetic field components of the modelled and analytical solutions.

.. _hertzian_dipole_fs_results_diffs:

.. figure:: ../../tests/analytical/hertzian_dipole_fs/hertzian_dipole_fs_vs_analytical_diffs.png
    :width: 600 px

    Percentage differences between the modelled and analytical solutions.

The match between the analytical and numerically modelled solutions is excellent. The maximum difference is approximately 1%, which is observed in the Ez field component (the same direction as the Hertzian dipole source). The other electric field components exhibit maximum differences of approximately 0.5%, and the magnetic field components 0.25%.


Half-wave dipole in free space
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See the :ref:`section on antenna example models <example-wire-dipole>` for the simulated s11 parameter and input impedance of a half-wave dipole antenna in free space. The resonant frequency and input impedance from the model agree very well with the theoretical predictions for a half-wave dipole antenna.