Calculated function expression
==================================


Data from the first group
-------------------------

**In the case of data from the first group**, with :math:`N` occurrences of the factor keyword COUR corresponding to :math:`N` time intervals, the arguments entered at the kth occurrence of the factor keyword are assigned the index :math:`(k)`, i.e.:

:math:`{I}_{1}^{(k)},{\tau }_{1}^{(k)},{\phi }_{1}^{(k)},{I}_{2}^{(k)},{\tau }_{2}^{(k)},{\phi }_{2}^{(k)},{t}_{1}^{(k)},{t}_{f}^{(k)}`


Full signal
~~~~~~~~~~~~~~~

The expression for the calculated function is:

:math:`\begin{array}{cc}F(t)=& {4.10}^{-7}{I}_{1}^{(k)}{I}_{2}^{(k)}\\ & \times \left[\mathrm{cos}(2\pi \mathrm{fr}(t-{t}_{r}^{(k)})+{\phi }_{1}^{(k)})-\mathrm{cos}{\phi }_{1}^{(k)}{e}^{-\frac{t-{t}_{r}^{(k)}}{{\tau }_{1}^{(k)}}}\right]\\ & \times \left[\mathrm{cos}(2\pi \mathrm{fr}(t-{t}_{r}^{(k)})+{\phi }_{2}^{(k)})-\mathrm{cos}{\phi }_{2}^{(k)}{e}^{-\frac{t-{t}_{r}^{(k)}}{{\tau }_{2}^{(k)}}}\right]\end{array}`


Continuous signal
~~~~~~~~~~~~~~~

:math:`F(t)={4.10}^{-7}{I}_{1}^{(k)}{I}_{2}^{(k)}\times \left[\frac{1}{2}\cdot \mathrm{cos}({\phi }_{2}^{(k)}-{\phi }_{1}^{(k)})+\mathrm{cos}{\phi }_{1}^{(k)}\times \mathrm{cos}{\phi }_{2}^{(k)}\times {e}^{-(\frac{1}{{\tau }_{1}^{(k)}}+\frac{1}{{\tau }_{2}^{(k)}})(t-{t}_{r}^{(k)})}\right]`

For :math:`t\in \left[{t}_{i}^{(k)},{t}_{f}^{(k)}\right],k=\mathrm{1,}N`

with :math:`{t}_{r}^{(k)}` defined by:

:math:`\begin{array}{cccccc}{t}_{r}^{(1)}& =& {t}_{i}^{(1)}& & & \\ {t}_{r}^{(k)}& =& {t}_{r}^{(k-1)}& \text{si}& {t}_{i}^{(k)}={t}_{f}^{(k-1)},& k=\mathrm{2,}N\\ {t}_{r}^{(k)}& =& {t}_{i}^{(k)}& \text{si}& {t}_{i}^{(k)}>{t}_{f}^{(k-1)},& k=\mathrm{2,}N\end{array}`


.. csv-table::

    ":math:`F(t)=0` ", "yes", ":math:`{t}_{i}^{(k)}>{t}_{f}^{(k-1)}` ", "and :math:`t\in \left[{t}_{f}^{(k-1)},{t}_{i}^{(k)}\right],k=\mathrm{2,}N`"
    "or", "yes", ":math:`t>{t}_{f}^{(N)}` ", ""



Data from the second group
------------------------

**In the case of data from the second group**, with an occurrence of the keyword COUR_PRIN (a main current) and :math:`N` occurrences of the keyword factor COUR_SECO (:math:`N` secondary currents), the arguments entered in the :math:`{k}^{\mathrm{ième}}` occurrence of COUR_SECO are assigned to the index :math:`(k)` i.e.: :math:`{I}_{2}^{(k)},{\tau }_{2}^{(k)},{\Phi }_{2}^{(k)},{d}^{(k)}` and possibly :math:`{I}_{2R}^{(k)},{\tau }_{2R}^{(k)},{\Phi }_{2R}^{(k)}`.


Case where :math:`t` belongs to the first current interval
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Full signal
^^^^^^^^^^^^^^

The expression for the calculated function then becomes: if :math:`t\in [{t}_{i},{t}_{f}]`

:math:`\begin{array}{ccc}F(t)={4.10}^{-7}& \times & {I}_{1}\cdot \left[\mathrm{cos}(2\pi \cdot \mathrm{fr}\cdot (t-{t}_{i})+{\Phi }_{1})-\mathrm{cos}({\Phi }_{1})\cdot {e}^{-\frac{t-{t}_{i}}{{\tau }_{1}}}\right]\\ & \times & \sum _{k=\mathrm{1,}N}\frac{{I}_{2}^{(k)}}{{d}^{(k)}}\left[\mathrm{cos}(2\pi \cdot \mathrm{fr}\cdot (t-{t}_{i})+{\Phi }_{2}^{(k)})-\mathrm{cos}({\Phi }_{2}^{(k)})\cdot {e}^{-\frac{t-{t}_{i}}{{\tau }_{2}^{(k)}}}\right]\end{array}`


Continuous signal
^^^^^^^^^^^^^^

:math:`\begin{array}{c}F(t)={4.10}^{-7}\cdot {I}_{1}\\ \times \sum _{k=\mathrm{1,}N}\frac{{I}_{2}^{(k)}}{{d}^{(k)}}\left[\frac{1}{2}\mathrm{cos}({\Phi }_{2}^{(k)}-{\Phi }_{1}^{(k)})+\mathrm{cos}({\Phi }_{1})\cdot \mathrm{cos}({\Phi }_{2}^{(k)})\cdot {e}^{-(\frac{1}{{\tau }_{1}}+\frac{1}{{\tau }_{2}^{(k)}})(t-{t}_{i})}\right]\end{array}`


Case where :math:`t` belongs to the second current interval (reclosing)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


Full signal
^^^^^^^^^^^^^^

The expression for the calculated function then becomes: if :math:`t\in [{t}_{\mathrm{iR}},{t}_{\mathrm{fR}}]`

:math:`\begin{array}{cc}F(t)={4.10}^{-7}\cdot {I}_{1R}& \left[\mathrm{cos}(2\pi {f}_{r}(t-{t}_{iR})+{\Phi }_{1R})-\mathrm{cos}({\Phi }_{1R})\cdot {e}^{-\frac{t-{t}_{iR}}{{\tau }_{1R}}}\right]\\ & \left[\mathrm{cos}(2\pi {f}_{r}(t-{t}_{iR})+{\Phi }_{2R}^{(k)})-\mathrm{cos}({\Phi }_{2R}^{(k)})\cdot {e}^{-\frac{t-{t}_{iR}}{{\tau }_{2R}^{(k)}}}\right]\end{array}`

NB: we must have :math:`{t}_{\mathrm{iR}}>{t}_{f}` and :math:`{t}_{\mathrm{iR}}\ne 0`


Continuous signal
^^^^^^^^^^^^^^

:math:`\begin{array}{}F(t)={4.10}^{-7}\cdot {I}_{1R}\\ \times \sum _{k=\mathrm{1,}N}\frac{{I}_{2R}^{(k)}}{{d}^{(k)}}\left[\frac{1}{2}\mathrm{cos}({\Phi }_{2R}^{(k)}-{\Phi }_{1R}^{(k)})+\mathrm{cos}({\Phi }_{1R})\cdot \mathrm{cos}({\Phi }_{2R}^{(k)})\cdot {e}^{-(\frac{1}{{\tau }_{1R}}+\frac{1}{{\tau }_{2R}^{(k)}})(t-{t}_{\mathrm{iR}})}\right]\end{array}`

In addition:

When reclosing: :math:`F(t)=0` if :math:`t\in [{t}_{f},{t}_{\mathrm{iR}}]` or :math:`t>{t}_{\mathrm{fR}}`

When there is no reclosing: :math:`F(t)=0` if :math:`t>{t}_{f}`