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 wye-connected. 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 so-called direct ( d ) axis. We also define a quadrature ( q ) axis that leads the d -axis by 90°. The angle between the d-axis and the a-phase of the stator winding is denoted by θ_d.

Fig. 1.15 Schematic diagram of a synchronous generator.

Let the self-inductance 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

	(1.82)
	(1.83)
	(1.84)

The self-inductance 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

	(1.89)
	(1.90)
	(1.91)

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 phase-a 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.,

	(1.95)
	(1.96)
	(1.97)

Substituting (1.92), (1.94), (1.95), (1.96) and (1.97) in (1.89) to (1.91) we get

	(1.98)
	(1.99)
	(1.100)

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..