# Electrical description of a synchronous machine

All three phases of the stator windings produce a main field that is connected to the rotor. The useful inductance Lh and the useful reactance Xh = 2nfjLh reflect this. Besides the useful field there are leakage fields that are not connected to the rotor. The leakage reactance Xa describes these leakages. The rotor field of a rotating rotor induces a voltage leak in the stator. This voltage is also called the synchronous internal voltage Vp. The rms value of the induced voltage is proportional to the excitation current IE in the rotor:

in other words, the induced voltage can be adjusted by changing the excitation current.

Besides the voltage drop over the reactances there is an ohmic voltage drop at the stator resistance Rr Hence, the equation for the equivalent circuit that describes the connection between the stator current Ij and stator voltage V1 for one phase becomes:

The influence of the stator resistance R1 is low for large machines so that it can be neglected. The useful reactance Xh and the leakage reactance Xa can be united as the synchronous reactance:

This simplifies the equation of the one-phase equivalent circuit (Figure 5.24):

Figure 5.24 Simple Equivalent Circuit (R = 0) of a Cylindrical Rotor Machine for One Phase

Figure 5.24 Simple Equivalent Circuit (R = 0) of a Cylindrical Rotor Machine for One Phase

A synchronous machine can generate reactive power if desired. This is the most significant advantage of the synchronous machine compared to the asynchronous machine described later. The stator current of a synchronous machine can run through all phase angles. This is also called four-quadrant operation. Figure 5.25 shows the vector diagrams of a cylindrical rotor machine for four different operation conditions.

Depending on the phase angle, states are described as underexcitation and overexcitation. If a synchronous machine is underexcited, it behaves like a coil and absorbs reactive current. An overexcited synchronous machine has the same behaviour as a capacitor and generates lagging reactive current.

Underexcitation-

Underexcitation-

Generator mode

Generator mode

Figure 5.25 Vector Diagrams of a Synchronous Machine with Cylindrical Rotor

Angular relations in the vector diagram provide the following relation:

Hence, the phase current I1 and the phase angle ç depend on load and excitation. The load angle & increases with rising load, i.e. torque M. The synchronous internal voltage Vp depends only on the excitation current. The electrical active power P1 of the stator

can be calculated from the active powers of the three winding phases of the stator. With the synchronous speed nS of the rotating field and the torque that is associated with the active power becomes:

According to this definition, the power P and torque M are negative in generator mode with negative load angle & and the machine produces power. The rotor torque is higher than the torque that corresponds to the active power because friction, gearbox and other losses reduce the rotor torque.

Figure 5.26 shows the curve of the torque M over the load angle A synchronous machine produces the theoretical maximum torque, called the pull-out torque, Mp, at load angles of ±n/2. The magnitude of this torque changes with the internal voltage and therefore with the excitation current. If the load torque of the machine increases above the pull-out torque Mp, the machine falls out of step. If it operates in motor mode it stops. If it operates in generator mode, the rotor runs faster than the rotating field in the stator and the machine overspeeds, in which case high centrifugal forces could destroy the rotor of a wind generator. Reliable safety systems such as aerodynamic brakes must avoid this critical operation mode.