Recently, engineering return stroke models were extended to take into account the presence of an elevated strike object, which was modeled as an ideal uniform transmission line. We show in this paper that the current distribution associated with these extended models exhibits a discontinuity at the return stroke wave front, which cannot be considered to be physically plausible. This discontinuity arises from the fact that the current injected into the tower from its top is reflected back and forth at its ends and portions of it are transmitted into the channel; these transmitted pulses, which are supposed to travel at the speed of light, catch up with the return stroke wave front traveling at a lower speed, and no current is allowed to flow in the leader region above the front. This discontinuity needs to be carefully treated when calculating the radiated electromagnetic field through an additional term in the electromagnetic field equations, the so-called "turn-on" term. We additionally derive a general analytical formula describing the turn-on term associated with this discontinuity for various engineering models. We present simulation results illustrating the effect of the turn-on term on the radiated electric and magnetic fields for an elevated strike object of height h = 168 m, corresponding to the Peissenberg tower in Germany. Finally, possible modifications to engineering models are suggested in order to eliminate this discontinuity