  We present a study of the acceleration phase of line-driven winds in AGNs, in
order to examine the physical conditions for the existence of such winds for a
wide variety of initial conditions. We built a simple and fast non-hydrodynamic
model, QWIND, where we assume that a wind is launched from the accretion disc
at supersonic velocities of the order of a few 10^2 km/s and we concentrate on
the subsequent supersonic phase. We show that this model can produce a wind
with terminal velocities of the order of 10^4 km/s. There are three zones in
the wind, only the middle one of which can launch a wind: in the inner zone the
wind is too ionized and so experiences only the Compton radiation force which
is not effective in accelerating gas. This inner failed wind however plays an
important role in shielding the next zone, lowering the ionization parameter
there. In the middle zone the lower ionization of the gas leads to a much
larger radiation force and the gas achieves escape velocity This middle zone is
quite thin (about 100 gravitational radii). The outer, third, zone is shielded
from the UV radiation by the central wind zone and so does not achieve a high
enough acceleration to reach escape velocity. We also describe a simple
analytic approximation of our model, based on neglecting the effects of gravity
during the acceleration phase. This analytic approach is in agreement with the
results of the numerical code, and is a powerful way to check whether a
radiation driven wind can be accelerated with a given set of initial
parameters. Our analytical analysis and the fast QWIND model are in agreement
with more complex hydrodynamical models, and allow an exploration of the
dependence of the wind properties for a wide set of initial parameters: black
hole mass, Eddington ratio, initial density profile, X-ray to UV ratio.
