This paper presents new contributions to the study of the positive discharge in large air gaps. A model enabling us to determine the voltage U50 and the k-factor in non-standard geometry using the circuit model developed by Beroual's group for long positive and negative discharges is presented. This model is based on an equivalent circuit diagram, the parameters of which vary with time according to the leader channel characteristics and the geometry/morphology of the discharge. The propagation of the leader is based on a criterion related to the calculation of the electric field at its head and that takes into account the randomness of the discharge path. As with most of models found in the literature, this model applies only to the point-plane type electrodes gap. This paper consists of two parts: (1) the first part, presents how to extend the proposed model to complex geometries such as rod–rod electrode gaps; and (2) the second part presents the experimental tests we achieved for rod-plane and rod–rod geometries. The experimental results especially the values of U50 and k-factor are compared with the simulated ones deduced from our model, for two types of voltage waveforms: the lightning impulse shape and the switching impulse shape (precisely, the critical impulse shape).