**Double Line to Ground Fault (LLG):**

Figure 11.9 shows a Double Line to Ground Fault at *F *in a power system. The fault may in general have an impedance Z^{f }as shown.

The current and voltage (to ground) conditions at the fault are expressed as

The symmetrical components of voltages are given by

from which it follows that

From Eqs. (11.22a) and (11.22b)

From Eqs. (11.19), (11.22a) and (11.23), we can draw the connection of sequence networks as shown in Figs. 11.10a and b. The reader may verify this by writing mesh and nodal equations for these figures.

In terms of the Thevenin equivalents, we can write from Fig. 11.10b Ea

The above result can be obtained analytically as follows:

Substituting for V_{a1,}V_{a2} and V_{a0} in terms of E_{a} in Eq. (11.1) and premultiplying both sides by Z^{-1} (inverse of sequence impedance matrix), we get

Premultiplying both sides by row matrix [1 1 1] and using Eqs. (11.19) and (11.20), we get

From Eq. (11.22a), we have

Substituting

or

Substituting this value of I_{a0} in Eq. (11.26) and simplifying, we finally get

If the fault takes place from loaded conditions, the positive sequence network will be modified as discussed in Sec. 11.3.