Types of Generator:

Generators can be classified into following types-

graph TD  
Generator  
Generator-->DC-generator  
DC-generator-->seperately-excited  
DC-generator-->self-excited --> compound-->long-shunt   
compound-->shortshunt  
self-excited --> shunt  
self-excited --> series  
Generator-->AC-generator  
AC-generator-->synchronous  
AC-generator-->induction

DC generator types:

1. Separately excited: In this type of generator a separate DC source is used to energize the field magnets i.e DC is provided to increase the strength of the magnetic fields produced by the magnets in the stator.
2. Self excited: In self excited generators the magnets are energized by own current produced. The current is produced due to the residual magnetism. Residual magnetism is the magnetism of a permanent magnet, which is the remnant magnetism after a magnet has been magnetised by an external field and the field has been removed. The voltage develop process of a self excited generator has been discussed in details here (Link). Self excited generators can further be classified into following category:
   1. Series wound generator: The properties of a series wound generator are-
      1. The field winding and the armature are connected in series.
      2. Same current flows through armature and load
      3. increasing load current increases the output voltage

Ia=IL=I

V=IRa

ii. Shunt wound generator: The properties of a series wound generator are-

armature and shunt field are connected in parallel

armature current and load voltage are inversely prooportional to each other

V = Terminal voltage

Eg = Generated EMF

Ish = Current flowing through the shunt field

Ia = Armature current

IL = Load current

Rsh= Shunt winding resistance

Ra = Armature resistance

                                                                           Ia = Ish + IL

                                                                          V = Eg − IaRa

iii. Compound Generator: Both series and shunt field wounding are combined to make a

    compound wound generator. Compound wound generator are of two types:

a) Long Shunt Compound DC Generators: In a long shunt-wound generator,  the shunt field winding is connected in parallel with both armature and series field winding.

From the figure,

Ia= Ise  

Ia= Ish+  IL

Ish =V/Rsh

V = Eg - IaRa - aRse

b) Short Shunt Compound DC Generators: In this type of generator, the shunt field winding is connected only across the armature.

Ia=Ise+ Ish

Ise= IL

Ia=IL+ Ish

Eg = V + IaRa + IseRse

Ise = IL

V=Eg - IaRa - ILRse