Section III Synchronous Machine Model
The schematic diagram of a synchronous generator is shown in Fig. 1.15. This contains three stator windings that are spatially distributed. It is assumed that the windings are wyeconnected. The winding currents are denoted by i_{a} , i_{b} and i_{c}. The rotor contains the field winding the current through which is denoted by i_{f} . The field winding is aligned with the socalled direct ( d ) axis. We also define a quadrature ( q ) axis that leads the d axis by 90°. The angle between the daxis and the aphase of the stator winding is denoted by θ_{d}.
Fig. 1.15 Schematic diagram of a synchronous generator.
Let the selfinductance of the stator windings be denoted by L_{aa}, L_{bb}, L_{cc} such that

(1.80) 
and the mutual inductance between the windings be denoted as

(1.81) 
The mutual inductances between the field coil and the stator windings vary as a function of θ_{d} and are given by
The selfinductance of the field coil is denoted by L_{ff}.
The flux linkage equations are then given by

(1.85) 

(1.86) 

(1.87) 

(1.88) 
For balanced operation we have
Hence the flux linkage equations for the stator windings (1.85) to (1.87) can be modified as
For steady state operation we can assume

(1.92) 
Also assuming that the rotor rotates at synchronous speed ω_{s} we obtain the following two equations

(1.93) 

(1.94) 
where θ_{d0} is the initial position of the field winding with respect to the phasea of the stator winding at time t = 0. The mutual inductance of the field winding with all the three stator windings will vary as a function of θ_{d}, i.e.,
Substituting (1.92), (1.94), (1.95), (1.96) and (1.97) in (1.89) to (1.91) we get
Since we assume balanced operation, we need to treat only one phase. Let the armature resistance of the generator be R . The generator terminal voltage is given by

(1.101) 
where the negative sign is used for generating mode of operation in which the current leaves the terminal. Substituting (1.98) in (1.101) we get

(1.102) 
The last term of (1.102) is the internal emf e_{a} that is given by

(1.103) 
where the rms magnitude E_{i} is proportional to the field current

(1.104) 
Since θ_{d0} is the position of the d axis at time t = 0, we define the position of the q axis at that instant as

(1.105) 
Therefore (1.94) can be rewritten as

(1.106) 
Substituting (1.105) in (1.103) we get

(1.107) 
Hence (1.102) can be written as

(1.108) 
Contd.. 