0.3 Chapter 4: introduction to rotating machines  (Page 2/9)

Figure 4.1 Schematic view of a simple, two-pole, single-phase synchronous generator.

Assume a sinusoidal distribution of magnetic flux in the air gap of the machine in Fig.4.1.

• The radial distribution of air-gap flux density B is shown in Fig. 4.2(a) as a function of the spatial angle $\theta$ around the rotor periphery.
• As the rotor rotates, the flux –linkages of the armature winding change with time and the resulting coil voltage will be sinusoidal in time as shown in Fig 4.2(b). The frequency in cycles per second (Hz) is the same as the speed of the rotor in revolutions in second (rps).
• A two-pole synchronous machine must revolve at 3600 rpm to produce a 60Hz voltage.
• Note the terms “rpm” and “rps”.

Figure 4.2 (a) Space distribution of flux density and (b) corresponding waveform of

the generated voltage for the single-phase generator of Fig. 4.1.

A great many synchronous machines have more than two poles. Fig 4.3 shows in schematic form a four-pole single-phase generator.

• The field coils are connected so that the poles are of alternate polarity.
• The armature winding consists of two coils $(a_1,-a_1)$ and $(a_2,-a_2)$ connected in series by their end connections.
• There are two complete wavelengths, or cycles, in the flux distribution around the periphery, as shown in Fig. 4.4.
• The generated voltage goes through two complete cycles per revolution of the rotor.
• The frequency in Hz is thus twice the speed in rps.

Figure 4.3 Schematic view of a simple, four-pole, single-phase synchronous generator.

Figure 4.4 Space distribution of the air-gap flux density in an idealized,

four-pole synchronous generator.

When a machine has more than two poles, it is convenient to concentrate on a single pair of poles and to express angles in electrical degrees or electrical radians rather than in physical units.

• One pair of poles equals 360 electrical degrees or 2 $\pi$ electrical radians.
• Since there are poles/2 wavelengths, or cycles, in one revolution, it follows that

${\theta }_{\text{ae}}=(\frac{\text{poles}}{2}){\theta }_a$ (4.1)

Where ${\theta }_{\text{ae}}$ is the angle in electrical units and ${\theta }_a$ is the spatial angle.

• The coil voltage of a multipole machine passes through a complete cycle every time a pair of poles sweeps by, or (poles/2) times each revolution. The electrical frequency $f_e$ of the voltage generated is therefore

$f_e=(\frac{\text{poles}}{2})\frac{n}{\text{60}}\text{Hz}$ (4.2)

where n is the mechanical speed in rpm.Note that ${\omega }_e=(\text{poles}/2){\omega }_m$

• • The rotors shown in Figs.4.1 and 4.3 have salient, or projecting, poles with concentrated windings. Fig.4.5 shows diagrammatically a nonsalient-pole, or cylindrical, rotor.
• The field winding is a two-pole distributed winding; the coil sides are distributed in multiple slots around the rotor periphery and arranged to produce an approximately sinusoidal distribution of radial air-gap flux.
• Most power systems in the world operate at frequencies of either 50 or 60 Hz.
• A salient-pole construction is characteristic of hydroelectric generators because hydraulic turbines operate at relatively low speeds, and hence a relatively large number of poles is required to produce the desired frequency.
• Steam turbines and gas turbines operate best at relatively high speeds, and turbine- driven alternators or turbine generators are commonly two- or four-pole cylindrical- rotor machines.