Overhead lines
Line inductance and reactance per phase (GMD/GMR)
- where
- phase spacings (m)
- geometric mean distance (m)
- inductance per phase (mH/km)
- conductor GMR (mm); 0.7788 r for a solid round conductor
- inductive reactance per phase (Ω/km)
- frequency (Hz)
Explain this equation
Gives the positive-sequence series inductance and 50 Hz reactance per phase of a transposed three-phase overhead line. Use it when building line impedance data for voltage drop, fault level or protection settings and no manufacturer or utility data is to hand.
D_m is the geometric mean of the three phase spacings: it stands in for an equilateral spacing that gives the same average flux linkage once the line is transposed. D_s (GMR) is the radius of a fictitious thin tube with the same total flux linkage as the real conductor, including flux inside the metal. For a solid round wire that works out at r·e^−¼ = 0.7788 r.
Inductance comes from the flux linking each conductor, which grows with the log of how far away the return currents are. 0.2 is µ0/2π expressed in mH/km; the 1000 brings D_m in metres to the same unit as GMR in millimetres. X_L = 2πfL, with ÷1000 converting mH to H.
Sense check: 11–132 kV wood-pole and lattice lines give about 1.0–1.3 mH/km, i.e. 0.3–0.4 Ω/km at 50 Hz. Because of the logarithm, doubling every spacing only adds 0.2·ln2 ≈ 0.14 mH/km.
Watch out: this is positive sequence only. Zero-sequence impedance, which sets earth fault current, depends on earth return and needs Carson’s method.
Positive-sequence series inductance and reactance per phase of a transposed three-phase single-circuit overhead line, from the conductor spacings and geometric mean radius.
Enter your values to see the result.
Good to know
GMD is the cube root of the three phase spacings. For stranded conductors (ACSR, AAAC) use the GMR from the manufacturer or BS EN 50182 data; it is typically 0.75–0.81 × outer radius. Enter 0 for a solid round conductor and 0.7788 r (= r·e^−¼) is used. For a bundle, enter the bundle equivalent GMR (e.g. √(GMR × s) for twin, ∛(GMR × s²) for triple). 11–132 kV lines usually give about 0.3–0.4 Ω/km at 50 Hz.
Common mistake
Using the outer radius in place of GMR for inductance (that is correct only for capacitance), or mixing feet and metres between spacing and GMR.
Limits
Assumes a fully transposed line, uniform current distribution in the GMR sense, negligible earth-return effect (positive sequence only) and 50/60 Hz internal inductance. Does not give zero-sequence impedance (use Carson’s equations) or account for skin effect beyond what the data-sheet GMR includes.
Sources
- Grainger, J.J. & Stevenson, W.D. Power System Analysis (McGraw-Hill), chapters 4–6: inductance, capacitance and current-voltage relations of a transmission line
- Glover, J.D., Overbye, T.J. & Sarma, M.S. Power System Analysis and Design (Cengage), chapters 4–5: transmission line parameters and steady-state operation
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.