4. Calculated function expression

4.1. Data from the first group

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

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

4.1.1. Full signal

The expression for the calculated function is:

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

4.1.2. Continuous signal

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

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

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

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

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

4.2. 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 \(N\) occurrences of the keyword factor COUR_SECO (\(N\) secondary currents), the arguments entered in the \({k}^{\mathrm{ième}}\) occurrence of COUR_SECO are assigned to the index \((k)\) i.e.: \({I}_{2}^{(k)},{\tau }_{2}^{(k)},{\Phi }_{2}^{(k)},{d}^{(k)}\) and possibly \({I}_{2R}^{(k)},{\tau }_{2R}^{(k)},{\Phi }_{2R}^{(k)}\).

4.2.1. Case where \(t\) belongs to the first current interval

4.2.1.1. Full signal

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

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

4.2.1.2. Continuous signal

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

4.2.2. Case where \(t\) belongs to the second current interval (reclosing)

4.2.2.1. Full signal

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

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

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

4.2.2.2. Continuous signal

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

In addition:

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

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