**Capacitance of a Three Phase Line Unsymmetrical Spacing:**

Figure 3.7 shows the three identical conductors of radius r of a Capacitance of a Three Phase Line Unsymmetrical Spacing. It is assumed that the line is fully transposed. As the conductors are rotated cyclically in the three sections of the transposition cycle, correspondingly three expressions can be written for V_{ab}. These expressions are:

For the first section of the transposition cycle

For the second section of the transposition cycle

For the third section of the transposition cycle

If the voltage drop along the line is neglected, V_{ab} is the same in each transposition cycle. On similar lines three such equations can be written for V_{bc} = V_{ab} ∠ –120°. Three more equations can be written equating to zero the summation of all line charges in each section of the transposition cycle. From these nine (independent) equations, it is possible to determine the nine unknown charges. The rigorous solution though possible is too involved.

With the usual spacing of conductors sufficient accuracy is obtained by assuming

This assumption of equal charge/unit length of a line in the three sections of the transposition cycle requires, on the other hand, three different values of V_{ab} designated as V_{ab1}, V_{ab2} and V_{ab3} in the three sections. The solution can be considerably simplified by taking V_{ab} as the average of these three voltages, i.e.

Adding Eqs. (3.18) and (3.19), we get

As per Eq. (3.12) for balanced three-phase voltages

Use of these relationships in Eq. (3.20) leads to

The capacitance of line to neutral of the transposed line is then given by

For air medium (k_{r} = 1)

It is obvious that for equilateral spacing D_{eq} = D, the above (approximate) formula gives the exact result presented earlier.

The line charging current for a three-phase line in phasor form is