Transmission line model of lightning return strokes generalized to include a tall grounded strike object and an upward connecting leader

We have proposed the transmission line (TL) model of lightning return strokes generalized to include a tall grounded strike object and an electrically long upward connecting leader. This model incorporates a lumped series voltage source at the junction between the descending leader and an upward connecting leader launched from the top of the strike object. We have derived expressions for current along the strike object and along the lightning channel (both above and below the leader junction point), as a function of the lightning short-circuit current, and also for other special cases such as lightning strikes to flat ground with or without an electrically long upward connecting leader. Regardless of the presence of a strike object and/or an upward connecting leader, the total charge transfer to ground is the same