Transients and insulation
Travelling wave reflection and transmission at a junction
- where
- reflection coefficient
- surge impedance beyond the junction (Ω)
- surge impedance of incoming line (Ω)
- transmission coefficient
- reflected voltage (kV)
- incident surge voltage (kV)
- transmitted voltage (kV)
Explain this equation
Gives the voltages reflected and transmitted when a travelling surge reaches a change in surge impedance, such as an overhead line meeting a cable, a transformer or an open breaker.
Z_1 is the surge impedance the wave is travelling on and Z_2 the impedance it meets. ρ is the fraction reflected back; τ the fraction passed on. V_i is the incident surge.
At the junction voltage must be continuous and the current must balance. Solving those two conditions for a wave arriving on Z_1 gives ρ = (Z_2 − Z_1)/(Z_2 + Z_1) and τ = 2Z_2/(Z_2 + Z_1), with τ = 1 + ρ.
Sense check: an open end (Z_2 very large) gives τ = 2, so voltage doubles. A short circuit gives ρ = −1. A 400 Ω line into a 40 Ω cable passes only about 18 % of the surge.
Watch out: in a short cable repeated reflections build the cable voltage back up towards the incident value, so a single reflection understates the stress.
Voltage reflected and transmitted when a travelling surge meets a change in surge impedance, such as an overhead line to cable junction.
Enter your values to see the result.
Good to know
τ = 1 + ρ. An open end (Z2 → ∞) doubles the voltage; a short circuit (Z2 = 0) reflects −1. Line-to-cable (Z2 ≪ Z1) gives a small transmitted voltage, but repeated reflections in a short cable build it back up, so surge arresters are still fitted at cable terminations. For several branches, enter their parallel surge impedance as Z2.
Common mistake
Applying the junction coefficients to the steady state, or forgetting multiple reflections in short cable sections (lattice diagram) which drive the cable end towards the incident value.
Limits
Lossless lines, resistive (real) surge impedances, first reflection only at a single discontinuity. Lumped capacitance or inductance at the junction, attenuation and corona distortion are not modelled.
Sources
- Naidu, M.S. & Kamaraju, V. High Voltage Engineering (McGraw-Hill Education)
- Kuffel, E., Zaengl, W.S. & Kuffel, J. High Voltage Engineering Fundamentals, 2nd ed. (Newnes, 2000)
- Grainger, J.J. & Stevenson, W.D. Power System Analysis (McGraw-Hill), chapters 4–6: inductance, capacitance and current-voltage relations of a transmission line
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.