  Conditions for net fast ion drive are derived for beam-driven, sub-cyclotron
compressional (CAE) and global (GAE) Alfv\'en eigenmodes, such as those
routinely observed in spherical tokamaks such as NSTX(-U) and MAST. Both co-
and counter-propagating CAEs and GAEs are investigated, driven by the ordinary
and anomalous Doppler-shifted cyclotron resonance with fast ions. Whereas prior
results were restricted to vanishingly narrow distributions in velocity space,
broad parameter regimes are identified in this work which enable an analytic
treatment for realistic fast ion distributions generated by neutral beam
injection. The simple, approximate conditions derived in these regimes for beam
distributions of realistic width compare well to the numerical evaluation of
the full analytic expressions for fast ion drive. Moreover, previous results in
the very narrow beam case are corrected and generalized to retain all terms in
$\omega/\omega_{ci}$ and $k_\parallel/k_\perp$, which are often assumed to be
small parameters but can significantly modify the conditions of drive and
damping when they are non-negligible. Favorable agreement is demonstrated
between the approximate stability criterion, simulation results, and a large
database of NSTX observations of cntr-GAEs.
