In fact, if the transformer is "perfect", the output will be
exactly zero.  Also it may burn up, unless the primary
DC resistance is very high.

Are you operating on 50 Hz or 60 Hz power?  A cheap..uh..
economical transformer designed for 60 Hz operation only
may saturate at 50 Hz and will accordingly run very hot.

> All this is true except You MUST properly phase the
> windings you connect in series or parallel.  if you fail to
> do this properly the fields will be bucking each other,
> the output low and the transformer will get quite hot.