Abstract: The electromagnetic coupling between multi-circuit transmission lines on the same tower necessitates optimization of the four-legged shunt reactors installed on them.This paper present the equations for calculating the secondary arc current as well as the recovery voltage through establishing general electromagnetic coupling model for multi-circuit transmission lines.The key factors influencing the secondary arc current and the recovery voltage were researched.A dynamic secondary arc model for ultra high voltage（UHV） transmission lines was built to investigate the impacts of the initial arc length,the neutral reactor on the arcing time of the secondary arc on multi-circuit transmission lines.Differences in optimal value of the neutral reactor of the four-legged shunt reactors between single-circuit and multi-circuit were discussed.Resonance frequency analysis was carried out regarding the resonant over-voltages caused by unbalanced switching operations as well as by electromagnetic induction between circuits.A simplified formula for calculating the inherent resonance points was deduced which indicated the range of the neutral reactor and also the method of its re-optimization that could be applied to the multi-circuit transmission lines.The results present useful reference for design of the four-legged shunt reactors on extra high voltage（EHV）/UHV multi-circuit transmission lines on the same tower.