Overhead lines
Ferranti voltage rise at the open receiving end
- where
- electrical length of the line (rad)
- frequency (Hz)
- series inductance per phase (mH/km)
- capacitance to neutral per phase (nF/km)
- line length (km)
- open-circuit receiving-end voltage (kV)
- sending-end line-to-line voltage (kV)
- approximate voltage rise (%)
Explain this equation
Gives the steady-state voltage at the open far end of a line energised from a stiff source: the Ferranti effect. Use it to check whether an unloaded line or cable will exceed its rated voltage and whether shunt reactors are needed.
βl is the electrical length of the line in radians: phase constant β = 2πf√(LC) per km, times the length l. With no load, charging current flows through the series inductance and the voltage rises along the line. V_s is the sending-end voltage held by the source; V_r is the open-end voltage.
For a lossless line the open-circuit voltage ratio is 1/cos(βl), from the hyperbolic long-line equations. Expanding the cosine for small angles gives the familiar rise of (βl)²/2, which is the ω²LCl²/2 approximation, shown as a percentage.
Sense check: an overhead line gains about 0.5 % at 100 km at 50 Hz, about 2 % at 200 km and 5 % at 300 km. The rise grows with the square of length, so doubling the length roughly quadruples it until the approximation breaks down.
Watch out: source reactance adds to the rise, and the simple approximation understates long lines; a quarter-wave line (about 1500 km at 50 Hz) gives unbounded voltage.
Steady-state voltage at the open (unloaded) receiving end of a line energised from a stiff source, from the lossless long-line equation.
Enter your values to see the result.
Good to know
The rise grows with the square of length: about 0.5 % at 100 km and 2 % at 200 km at 50 Hz for overhead lines, much more for cables. The approximation (βl)²/2 is good below about 300 km. The source impedance adds further rise because the charging current flows through it.
Common mistake
Ignoring the source (and transformer) reactance, which adds to the Ferranti rise, or using the approximation for very long lines or cable circuits.
Limits
Lossless distributed line, infinite (zero-impedance) source, no shunt reactors, sinusoidal steady state. Transient energisation overvoltages are higher; see IEC 60071-2 for switching overvoltages.
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
- Naidu, M.S. & Kamaraju, V. High Voltage Engineering (McGraw-Hill Education)
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.