# Chapter 5: synchronous machines

Chapter 5: Synchronous Machines

This lecture note is based on the textbook # 1. Electric Machinery - A.E. Fitzgerald, Charles Kingsley, Jr., Stephen D. Umans- 6th edition- Mc Graw Hill series in Electrical Engineering. Power and Energy

• Main features of synchronous machines:
• A synchronous machine is an ac machine whose speed under steady-state conditions is proportional to the frequency of the current in its armature.
• The rotor, along with the magnetic field created by the dc field current on the rotor, rotates at the same speed as, or in synchronism with, the rotating magnetic field produced by the armature currents, and a steady torque results.

Figure 5.1 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and

(c) Y connection of the windings.

§5.1 Introduction to Polyphase Synchronous Machines

• Synchronous machines:
• Armature winding: on the stator, alternating current.
• Field winding: on the rotor, dc power supplied by the excitation system.
• Cylindrical rotor: for two- and four-pole turbine generators.
• Salient-pole rotor: for multipolar, slow-speed, hydroelectric generators and for most synchronous motors.
• Acting as a voltage source:
• Frequency determined by the speed of its mechanical drive (or prime mover).
• The amplitude of the generated voltage is proportional to the frequency and the field current.

$\begin{array}{}{\lambda }_{a}={k}_{w}{N}_{\text{ph}}{\Phi }_{p}\text{cos}\left(\left(\frac{\text{poles}}{2}\right){\omega }_{m}t\right)\\ \text{}={k}_{w}{N}_{\text{ph}}{\Phi }_{p}\text{cos}{\omega }_{\text{me}}t\end{array}$ (5.1)

${\omega }_{\text{me}}=\left(\frac{\text{poles}}{2}\right){\omega }_{m}$ (5.2)

${e}_{a}=\frac{{\mathrm{d\lambda }}_{a}}{\text{dt}}={k}_{w}{N}_{\text{ph}}\frac{{\mathrm{d\Phi }}_{p}}{\text{dt}}\text{cos}{\omega }_{\text{me}}t-{\omega }_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}\text{sin}{\omega }_{\text{me}}t$ (5.3)

${e}_{a}=-{\omega }_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}\text{sin}{\omega }_{\text{me}}t$ (5.4)

${E}_{\text{max}}={\omega }_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}={2\pi f}_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}$ (5.5)

${E}_{\text{rms}}=\frac{2\pi }{\sqrt{2}}{f}_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}=\sqrt{2}{\mathrm{\pi f}}_{\text{me}}{k}_{w}{N}_{\text{ph}}{\Phi }_{p}$ (5.6)

• Synchronous generators can be readily operated in parallel: interconnected power systems.
• When a synchronous generator is connected to a large interconnected system containing many other synchronous generators, the voltage and frequency at its armature terminals are substantially fixed by the system.
• It is often useful, when studying the behavior of an individual generator or group of generators, to represent the remainder of the system as a constant-frequency, constant-voltage source, commonly referred to as an infinite bus.
• Analysis of a synchronous machine connected to an infinite bus.
• Torque equation:

$T\text{==}\frac{\pi }{2}\left(\frac{\text{poles}}{2}{\right)}^{2}{\Phi }_{R}{F}_{f}\text{sin}{\delta }_{\text{RF}}$ (5.7)

where

${\Phi }_{R}=$ resultant air-gap flux per pole

${F}_{f}=$ mmf of the dc field winding

${\delta }_{\text{RF}}=$ electric phase angle between magnetic axes of ${\Phi }_{R}$ and ${F}_{f}$

• The minus sign indicates that the electromechanical torque acts in the direction to bring the interacting fields into alignment.
• In a generator, the prime-mover torque acts in the direction of rotation of the rotor, and the electromechanical torque opposes rotation. The rotor mmf wave leads the resultant air-gap flux.
• In a motor, the electromechanical torque is in the direction of rotation, in opposition to the retarding torque of the mechanical load on the shaft.
• Torque-angle curve: Fig. 5.2.

Figure 5.2 Torque-angle characteristics.

• An increase in prime-mover torque will result in a corresponding increase in the torque angle.
• $T={T}_{\text{max}}$ : pull-out torque at $\delta =\text{90}$ .Any further increase in prime-mover torque cannot be balanced by a corresponding increase in synchronous electromechanical torque, with the result that synchronism will no longer be maintained and the rotor will speed up. $⇒$ loss of synchronism, pulling out of step.

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