The equivalent upstream network diagram during a solid short circuit is shown in Figure A-1.

Figure A-1: equivalent diagram upon occurrence of a solid short circuit

It is assumed that at the moment the short circuit occurs, the voltage phase is ?.

The current i( t) is determined by the following differential equation:

We shall solve this differential equation using Laplace transforms.

As a Laplace transform, the differential equation becomes:

as it is assumed that the current is zero (negligible) before the fault, then i( t =0) = 0.

The Laplace transform tables give us:

To reverse the Laplace transform I ( s), it must be written differently.

We know that we can write I( s) as:

We shall thus look for the value of coefficients A, B and C.

Search for coefficient C

By multiplying the two terms of the equation by , and by taking we obtain:

Let us take: Z ² = R ² + L ² ? ²

If ? is the network's natural phase displacement, we have: and

Search for coefficient A

If we make s stretch to infinity, we obtain the following equation:

hence:

Search for coefficient B

If we make s stretch to 0, we obtain the following equation:

We can thus write I ( s) as follows:

Using the Laplace transform tables, we can deduce i( t):