Synchronous Impedance and Synchronous Reactance

Synchronous Reactance :

     The armature reaction reactance and the leakage reactance present in a synchronous machine and acting at a same time. The two reactances are combined together and the sum is called the Synchronous reactance (Xs).

    Zs  =  (Ra)2 + (Xs)2

    Eo  =  No-Load emf

           =   ( V Cos φ + I*Ra )2  +  ( V Sin φ + I*Xs )2

     E  =  emf induced on load

It is vectorially less than Eo by I*Xa. Sometimes it is also written as Ea.

    V  =  Terminal voltage. It is vectorially less than Eo by I*Zs.

     I  =  Armature current / phase

    φ  =  Load p.f. angle

    Z  =  (Ra)2 + (XL)2

    Zs  =  (Ra)2 + (XL)2 

     As the load on the alternator is increased, the terminal voltage changes ( for constant excitation ) due to the following reasons :

(i) voltage drop due to armature resistance, I*Ra.

(ii) voltage drop due to armature leakage reactance,  I*XL.

(iii) voltage drop due to armature reaction.

     The voltage drop due to armature reaction may be accounted for by assuming the presence of a fictitious reactance Xa in the armature winding. The value of Xa is such that I*Xa of leakage reactance XL and ( fictitious ) armature reactance Xa is known as synchronous reactance Xs.


Xs  = XL + Xa


The voltage drop in an alternator under load is

=  I*Ra + j I*Xs

=  I ( Ra + j Xs )

   Xs  =  Synchronous reactance

Synchronous Impedance :

     Synchronous impedance may be defined as the vector sum of the armature resistance and synchronous impedance. 

     When an alternator is loaded, there will be voltage drop due to Ia  Ra which is in phase with Iand due to I Xs, which is leading I by 90°. The difference between the terminal voltage V is due to the induced emf E and drops in the resistance and reactance, Ia  Ra  and I Xs, The relationship between induced emf E and the terminal voltage V can be represented as


E  =  V + Ia Ra + jIa ( XL + Xa )

E  =  V + Ia Ra + jIa Xs

E  =  V + Ia ( Ra + jXs )

E  =  V + Ia Zs

The vector sum of the Ra and Xs is called synchronous impedance, Z.

Effect of Synchronous Impedance :

The following are the various effects of synchronous impedance :

1. Voltage drop increases.

2. Supply voltage decreases.

3. Power loss takes place ( since impedance includes resistance ).

4. Heat is produced.

5. Voltage regulation increases.

6. Efficiency decreases.

7. For large alternators cooling system is required.

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