On the basis of a distributed-source representation of the lightning channel, the mathematical formulations of the so-called engineering lightning return stroke models are generalized to take into account the presence of a vertically extended strike object. The strike object is modeled as a lossless uniform transmission line, and the reflection coefficients are all assumed to be constant. The distribution of current along the lightning channel for each model is expressed in terms of the "undisturbed" current, object height, and current reflection coefficients at the top and the bottom of the object. The undisturbed current is defined as the current that would flow in the channel if the current reflection coefficients at the extremities of the strike object were equal to zero, that is, the characteristic impedances of the lightning channel and the strike object were equal to each other and equal to the grounding impedance of the strike object. The distributed-source representation of the lightning channel adopted in this study allows for a more general and straightforward formulations of the generalized return-stroke models than the traditional representations implying a lumped current source at the bottom of the channel, including a self-consistent treatment of the impedance discontinuity at the tower top