SECTION 3: INDUCTANCE OF THREE-PHASE TRANSMISSION LINES

1. Inductance of a Three-Phase Line

A three-phase transmission line consists of three conductors carrying currents Ia, Ib, and Ic. For a balanced system, the currents have equal magnitudes and are displaced by 120°electrically:

                                                                             Ia + Ib + Ic = 0

The magnetic field produced by each phase links with the other phase conductors. Therefore, the inductance of each phase depends not only on its own conductor, but also on the currents and positions of the other conductors.

For a solid round conductor, the effective radius used in inductance calculations is:

                                                                             Ds = r = 0.7788R

where is the physical conductor radius Ds is the self-GMD or GMR of the conductor.

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2. Geometric Mean Distance (GMD)

Geometric mean distance represents the equivalent spacing between the phase conductors. For three conductors with mutual distances Dab, Dbc, and , Dca the equivalent GMD is:

                                                                               Dm = ∛DabDbcDca

Where:

• Dab= distance between phases and

• Dbc= distance between phases and

• Dca= distance between phases and

• Dm= equivalent GMD (m)

GMD accounts for the mutual spacing of the conductors in a three-phase line.

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3. Symmetrical Conductor Spacing

In a symmetrically spaced three-phase line, the three conductors form an equilateral triangle. All phase-to-phase distances are equal:

                                                                              Dab = Dbc = Dca = D

Therefore,

                                                                               Dm = ∛D³ = D

For a transposed line, the inductance per phase per unit length is:

                                                                               L = 2× 10⁻⁷ In ln (Dm/Ds) H/m

For symmetrical spacing, Dm = D , so: 

                                                                               L = 2× 10⁻⁷ In ln (D/Ds) H/m

This formula gives the inductance of one phase conductor, not the sum of all three phases.

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4. Unsymmetrical Conductor Spacing

In an unsymmetrically spaced three-phase line, the conductors do not form an equilateral triangle. Thus,

                                                                                Dab ≠ Dbc ≠ Dca 

The inductances of the individual phases may be unequal because the distances between the conductors differ. For a properly transposed line, each phase occupies each physical position for one-third of the line length. The average inductance of each phase can then be represented using the equivalent GMD:

                                                                                  Dm = ∛DabDbcDca

Hence, the average inductance per phase is:

                                                                                Lavg = 2× 10⁻⁷ In ln (Dm/Ds) H/m

For a solid round conductor, Ds = 0.7788R.

Therefore, 

                                                                              Lavg = 2× 10⁻⁷ In ln (∛DabDbcDca / 0.7788R)

The distances and radius must be expressed in the same units.

5. Transposition of Three-Phase Lines

Transposition is the periodic interchange of the physical positions of the three phase conductors along a transmission line. In a complete transposition cycle, each phase occupies each of the three positions for an equal length.

Each phase occupies all three positions over one complete cycle.

The average inductance is the arithmetic mean of the inductances associated with the three conductor positions:

                                                                         Lavg = (La + Lb + Lc) /  3

For a transposed three-phase line, this average can be expressed using GMD and GMR:

                                                                         Lavg = 2× 10⁻⁷ In ln (Dm/Ds) H/m

Transposition equalizes the average inductance of the phases in the idealized line model. It can also reduce the imbalance in induced voltages and the interference caused by unequal conductor positions.

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6. Important Equations — Quick Revision

Quantity                                                                                              Equation

GMR of a solid conductor                                                             Ds = 0.

GMD for three phases                                                                     Dm = ∛D

Inductance per phase                                                                      L = 2× 10⁻⁷ In ln (Dm/Ds) H/m

H/m

GMD for symmetrical spacing                                                     Dm = D

Average inductance after transposition                                    L = (La + Lb + Lc) /  3

Inductance for symmetrical spacing                                          L = 2× 10⁻⁷ In ln (D/Ds) H/m

H/m