The Equivalent Circuit of a Synchronous Generator

THE EQUIVALENT CIRCUIT OF A SYNCHRONOUS GENERATOR

The internal generated voltage produced in one phase of a synchronous generator can be called as E_A.  If the machine is not connected to a load (no armature current flowing), the terminal voltage will be equivalent to the voltage induced at the stator coils. This is due to the fact that there are no current flow in the stator coils hence no losses. When there is a load connected to the generator, there will be differences between E_A and V_f. These differences are due to:

a)      Distortion of the air gap magnetic field by the current flowing in the stator called armature reaction.

b)      Self inductance of the armature coil

c)      Resistance of the armature coils

d)      The effect of salient pole rotor shapes.

We will explore factors a, b, and c and derive a machine model from them.  The effect of salient pole rotor shape will be ignored, and all machines in this chapter are assumed to have non salient or cylindrical rotors. 

Armature Reaction

When the rotor is spun, a voltage E_A is induced in the stator windings.  If a load is attached to the terminals of the generator, a current flows.  But a 3-phase stator current flow will produce a magnetic field of its own.  This stator magnetic field will distorts the original rotor magnetic field, changing the resulting phase voltage.  This effect is called armature reaction because the armature (stator) current affects the magnetic field, which produced it in the first place. 

Refer to the diagrams below, showing a two-pole rotor spinning inside a 3-phase stator.

•  A rotating magnetic field produces the internal generated voltage E_A.
• The resulting voltage produces a lagging current flow when connected to a lagging load.
•  The stator current produces its own magnetic field B_S which produces its own E_stat in the stator windings.
• The field B_S adds to B_R distorting it into B_net.  The voltage E_stat adds to E_A, producing V_f at the output of the phase.
  
  
  (a)    There is no load connected to the stator. The rotor magnetic field B_R produces an internal generated voltage E_A whose peak coincides with direction of B_R.  With no load, there is no armature current and E_A will be equal to the phase voltage V_f.   
  (b)    When a lagging load is connected, the peak current will occur at an angle behind the peak voltage.
  (c)    The current flowing in the stator windings produces a magnetic field of its own.  This stator magnetic field B_S and its direction are given by the right-hand rule.  The stator field produces a voltage of its own called E_stat. 
  (d)    With 2 voltages and 2 magnetic fields present in the stator windings, the total voltage and the net magnetic field are:

How can the effects of armature reaction on the phase voltage be modeled?

-          The voltage E_stat lies at an angle of 90° behind the plane of I_A. 

-          The voltage E_stat is directly proportional to the current I_A. 

If X is a constant of proportionality, then the armature reaction voltage can be expressed as:

Therefore:

Thus, the armature reaction voltage can be modeled as an inductor in series with the internal generated voltage.

Self-inductance and Resistance of the Armature Coils

If the stator self-inductance is called L_A (reactance is X_A) while the stator resistance is called R_A, then the total difference between E_A and V_f is:

Where X_S = X + X_A,

The full equivalent circuit is shown below:

A dc power source is supplying the rotor field circuit, which is modeled by the coil’s inductance and resistance in series.  In series with R_F is an adjustable resistor R_adj which controls the flow of the field current.  The rest of the equivalent circuit consists of the models for each phase.  Each phase has an internal generated voltage with a series inductance X_S (consisting of the sum of the armature reactance and the coil’s self-inductance) and a series resistance R_A. 

If the 3 phases are connected in Y or ∆, the terminal voltage may be found as follows:

                                         

                                          

Ideally, the terminal voltage for all 3 phases should be identical since we assume that the load connected is balanced. If it is not balanced, a more in-depth technique is required.

The per-phase equivalent circuit: