The article provides analytical expressions for the electromagnetic fields generated by a lightning return stroke characterized by a channel-base current with arbitrary time waveform, in presence of either a perfectly conducting or a lossy ground, assuming the transmission line model for the current along the channel. In this second case, a time-domain analytical expression for the Cooray–Rubinstein formula is presented. The main idea that leads to the derivation of analytical formulas consists of dividing the channel into intervals in which the distance between the field source point and the observation point can be approximated with a linear function of the time, and of the spatial coordinates of both points. In the companion aticle, a detailed comparison is proposed with the classical (numerical) approach highlighting an excellent agreement both at close and far distances, considering all the values of practical interest for the ground conductivity. Moreover, the method guarantees a meaningful improvement in computational performance.