Overhead Line Parameters: Inductance, Capacitance and Reactance from GMD and GMR
Find the inductance, capacitance, reactance and charging current of a three-phase overhead line from the conductor size, the bundle and the spacing.
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How this calculator works
Enter the radius of one conductor, the number of conductors in each bundle and the spacing between them, then the three phase-to-phase spacings. The calculator finds the geometric mean distance between the phases (GMD) and the geometric mean radius of the bundle (GMR).
It gives the inductance L = 2e-7 ln(GMD/GMR) per metre and the capacitance C = 2 pi e0 / ln(GMD/r) per metre, converted to mH/km and nF/km, then the reactance and susceptance at your frequency.
With the line length and voltage it also gives the charging current and reactive power the line makes, and the surge impedance. Phases are assumed transposed.
Formulas used
- GMD = \sqrt[3]{D_{12}D_{23}D_{31}}
- L = 2\times10^{-7}\ln\frac{GMD}{GMR}\ \ \mathrm{H/m}
- C = \frac{2\pi\varepsilon_0}{\ln(GMD/r_b)}\ \ \mathrm{F/m}
- r' = 0.7788\,r
Worked example
With the default numbers in the calculator above:
- Geometric mean distance between the phases: 10.08 m
- Self geometric mean radius of one conductor (solid round): 7.79 mm
- Inductance per phase per km: 1.43 mH/km
- Capacitance to neutral per km: 8.04 nF/km
- Inductive reactance per km: 0.45 ohm/km
- Shunt susceptance per km: 2.53 uS/km
- Whole line: reactance and susceptance: 45.02 ohm and 252.72 uS
- Charging current and reactive power of the line: 32.10 A and 12.23 MVAr
- Surge impedance (lossless): 422.08 ohm
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
Why use bundled conductors?
A bundle behaves like a conductor of larger radius. It cuts the inductance, raises the capacitance and lowers the electric field at the surface, which reduces corona loss and radio noise.
Why is GMR 0.7788 times the radius?
The magnetic flux inside a solid round conductor adds an extra internal inductance that is the same as that of a conductor of radius r e to the minus one quarter, which is 0.7788 r.
What are typical values?
About 1.0 to 1.4 mH/km and 8 to 12 nF/km for single circuit overhead lines, which is a reactance of about 0.3 to 0.45 ohm/km at 50 Hz.
Does it include the effect of the earth?
No. For capacitance the earth effect is small for a high line. Transposed phases are assumed.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Grainger and Stevenson, Power System Analysis (inductance and capacitance of transmission lines); Weedy, Electric Power Systems.
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