Can you explain why there is an armature voltage even then the field current is zero in case of separately-excited dc generator?

if this happened it will be due to residual magentism and as soon as this magentism vanishes no voltage will be available , what iam saying can be interpreted mathematically

first the generated emf across the armature is given by

E= k* flux* speed

it is clear due to this equation the emf is directly prportional to the flux which means if the flux is reduced so does the emf and the revers is true, the reduction may be lead to zero flux value and thus the emf (zero field current)

secondly this emf covers the terminal voltage and the drop in armature winding if other possiple drops are ignored as

E= terminal voltage + armature drop

in this equation as E drops down as a result of reducing the flux and which in turn because of reduction in the field current , then right hand components will also drop down until both get zero values , so theoreyically no voltage will be exist at generator terminals when the field current comes to zero value, so the persistance of voltage is due to the residual field and will not last long