Synchronous Impedance

The actual generated voltage consists of the summation of two component voltages. One of these components voltages is the voltage that would be generated if there were no armature reaction. it is the voltage that would be generated because of only field excitation. This component of the generated voltage is called the excitation voltage, Eexc.

The other component of the generated voltage is called armature reaction voltage , EAR. This is the voltage that must be added to the excitation voltage to take care of the effect of armature reaction upon the generated voltage .

              Ea = Eexc + EAR

Since armature reaction results, in a voltage effect in circuit caused bu change in flux by current in the same circuit, its effect is of the nature of an inductive reactance .therefore EAR is equivalent to a voltage of inductive reactance and

            EAR  =  - j XAR  Ia

The inductive Reactance XAR is a fictitious reactance which will result in a voltage in armature circuit to account for the effect of armature reaction upon the voltage relations of the armature circuit. Therefore armature reaction voltage can be modelled as in inductor in series with internal generated voltage .

In additionato the effect of armature reaction , the stator winding also has a self inductance and resistance.

Let         La  = self inductance of stator winding.

             Xa  = self reactance of stator winding.

             Ra  = armature (stator) resistance.

The terminal voltage V is given by 

               V =  Ea - j XAR Ia - j Xa Ia - RaIa

where 

             RaIa = armature  resistance drop

            Xa Ia = armature leakage reactance drop

           XAR Ia = armature reaction voltage 

  

the armature reaction effect and the leakage flux effect in the machine are both represented by inductive reactance . therefore, it is customary to combine them into a single reactance, is called the synchronous reactance of the machine, Xs. 

          

                Xs = Xa + XAR

                 V =  Ea - j Xs Ia - RaIa

                 V =  Ea - (  Ra +j Xs) Ia

                 V =  Ea - Zs Ia

                Zs = Ra +j Xs

The impedance Zs is called synchronous impedance. 
   
The synchronous  Reactance Xs is the fictitious reactance employed to account for voltage effects  in the armature circuit produced by the actual armature leakage reactance and by the change in air gap flux caused by the armature reaction. 

similarly ,synchronous impedance  Zs is a fictitious impedance employed to account for voltage effects  in the armature circuit produced by the actual armature resistance, the actual armature leakage reactance, and the change in air gap flux produced by armature reaction.