Overhead lines
Line capacitance and charging current per phase
- where
- phase spacings (m)
- geometric mean distance (m)
- capacitance to neutral per phase (nF/km)
- permittivity of free space, 8.854 × 10⁻¹² F/m
- conductor outer radius (mm)
- charging current per phase (A/km)
- frequency (Hz)
- line-to-line voltage (kV)
- three-phase charging reactive power (Mvar/km)
Explain this equation
Gives the capacitance to neutral per phase of a transposed overhead line, and the charging current and reactive power it draws per kilometre. Useful for open-end voltage rise, reactive compensation and checking capacitive switching duties.
D_m is the geometric mean spacing, as for inductance. Here r is the actual outer radius, because the charge sits on the conductor surface and there is no internal-flux term. ε0 is the permittivity of free space. I_c uses the phase voltage V/√3 because each phase charges to neutral; Q_c is three times the phase value, which simplifies to 2πfCV².
The potential between a charged line and its neighbours rises with the log of spacing over radius, so C = 2πε0/ln(D_m/r). 2πε0 × 10¹² is 55.63, giving nF/km. The 10⁻⁶ and 10⁻⁹ factors tidy up nF and kV into A and Mvar.
Sense check: overhead lines run about 8–12 nF/km. A 132 kV line at 9 nF/km draws roughly 0.2 A/km and about 0.05 Mvar/km. Cables are 20 to 50 times higher.
Watch out: the earth and earth wires raise the true value by a few per cent; this simple form ignores them.
Capacitance to neutral per phase of a transposed three-phase overhead line, with the charging current and charging reactive power per kilometre.
Enter your values to see the result.
Good to know
Uses the actual outer radius (not GMR). Typical overhead lines are 8–12 nF/km. Charging is small at 11–33 kV but matters at 132 kV and above for long lines, open-end voltage rise and reactive compensation. For a bundle use the bundle equivalent radius √(r·s), ∛(r·s²) etc.
Common mistake
Using GMR instead of outer radius, or quoting Ic with the line-to-line voltage instead of phase voltage.
Limits
Ignores the effect of earth and earth wires (the method of images raises C by a few per cent for low lines), and assumes full transposition. Positive-sequence only.
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.