Q2Power System - I

Question

Q.2. Derive the expression for the capacitance of three phase unsymmetrical spaced transposed overhead transmission line.

Answer 1

For an unsymmetrically spaced 3-phase line, the conductor spacings Dab, Dbc, Dca are all different, which would cause unbalanced capacitance in each phase; transposition (cyclically interchanging conductor positions along the line so each conductor occupies every position for an equal length) restores a balanced (though not identical position-by-position) average capacitance to each phase over the full length of the line.

Consider phase 'a' conductor with charge +qa (and phases b, c similarly with qb, qc, where in a balanced system qa+qb+qc=0). The potential of conductor a due to the charges on all three conductors, averaged over the three transposition sections, can be derived using the standard method of averaging the potential difference formula in each of the three transposition cycles. For section 1 (conductor a in position 1, b in position 2, c in position 3): potential of conductor a relative to a remote reference is:

V_{a 1} = \frac{1}{2 π ε _{0}} [q_{a} ln \frac{1}{r} + q_{b} ln \frac{1}{D _{ab}} + q_{c} ln \frac{1}{D _{c a}}]

Similarly for section 2 (conductor a in position 2) and section 3 (conductor a in position 3), with the distances cyclically permuted. Averaging the three sections' expressions for Va (i.e., Va = (Va1+Va2+Va3)/3) and using qa+qb+qc=0 to eliminate qc = -(qa+qb) — or more directly, exploiting the fact that the average of ln(Dab)+ln(Dbc)+ln(Dca) over the three permutations naturally produces the GMD term — gives, after simplification:

V_{a} = \frac{1}{2 π ε _{0}} [q_{a} ln \frac{GM D}{r}], GM D = (D_{ab} D_{b c} D_{c a})^{1/3}

where the mutual terms from qb and qc combine, using qb+qc=-qa, into a single term involving the geometric mean of the three spacings, exactly analogous to the way GMD replaces the conductor spacing in the corresponding inductance formula for an unsymmetrically spaced, transposed line. The capacitance to neutral is then simply Ca = qa/Va:

C_{n} = \frac{2 π ε _{0}}{ln ( GM D / r )} F/m

This result shows that an unsymmetrically spaced but fully transposed line behaves, for capacitance calculation purposes, exactly like an equivalent symmetrically spaced line whose uniform spacing equals the geometric mean distance GMD = (Dab·Dbc·Dca)^(1/3) of the actual unsymmetrical spacing — this is precisely why GMD is such a fundamental and widely used parameter in transmission line parameter calculations, allowing the same simple formulas developed for symmetrical spacing to be applied directly to any transposed line configuration by substituting GMD for the conductor spacing.

Q2Power System - I

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