4.1 Problems - chapter 4

PROBLEMS

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

4.1 The rotor of a six-pole synchronous generator is rotating at a mechanical speed of 1200 r/min.

a. Express this mechanical speed in radians per second.

b. What is the frequency of the generated voltage in hertz and in radians per second?

c. What mechanical speed in revolutions per minute would be required to generate voltage at a frequency of 50 Hz?

4.2 The voltage generated in one phase of an unloaded three-phase synchronous generator is of the form v(t) = $V_0\text{cos}\mathrm{\omega t}$ . Write expressions for the voltage in the remaining two phases.

4.3 A three-phase motor is used to drive a pump. It is observed (by the use of a stroboscope) that the motor speed decreases from 898 r/min when the pump is unloaded to 830 r/min as the pump is loaded.

a. Is this a synchronous or an induction motor?

b. Estimate the frequency of the applied armature voltage in hertz.

c. How many poles does this motor have?

4.4 A three-phase Y-connected ac machine is initially operating under balanced three-phase conditions when one of the phase windings becomes open-circuited. Because there is no neutral connection on the winding, this requires that the currents in the remaining two windings become equal and opposite. Under this condition, calculate the relative magnitudes of the resultant positive- and negative-traveling mmf waves.

4.5 What is the effect on the rotating mmf and flux waves of a three-phase winding produced by balanced-three-phase currents if two of the phase connections are interchanges?

4.6 In a balanced two-phase machine, the two windings are displaced 90 electrical degrees in space, and the currents in the two windings are phase-displaced 90 electrical degrees in time. For such a machine, carry out the process leading to an equation for the rotating mmf wave corresponding to Eq.4.39 (which is derived for a three-phase machine).

4.7 This problem investigates the advantages of short-pitching the stator coils of an ac machine. Figure 4.1a shows a single full-pitch coil in a two-pole machine. Figure 4.1b shows a fractional-pitch coil for which the coil sides are β radians apart, rather than π radians ( ${\text{180}}^0$ ) as is the case for the full-pitch coil.

For an air-gap radial flux distribution of the form

$B_r=\sum _{\text{n odd}}{B}_n\text{cos}\mathrm{n\theta }$

where n = 1 corresponds to the fundamental space harmonic, n = 3 to the third space harmonic, and so on, the flux linkage of each coil is the integral of $B_r$ over the surface spanned by that coil. Thus for the nth space harmonic, the ratio of the maximum fractional-pitch coil flux linkage to that of the full-pitch coil is

$\frac{{\int }_{-\beta /2}^{\beta /2}{B}_n\text{cos}\mathrm{n\theta }\mathrm{d\theta }}{{\int }_{-\pi /2}^{\pi /2}{B}_n\text{cos}\mathrm{n\theta }\mathrm{d\theta }}=\frac{{\int }_{-\beta /2}^{\beta /2}\text{cos}\mathrm{n\theta }\mathrm{d\theta }}{{\int }_{-\pi /2}^{\pi /2}\text{cos}\mathrm{n\theta }\mathrm{d\theta }}=\mid \text{sin}(\mathrm{n\beta }/2)\mid$

It is common, for example, to fractional-pitch the coils of an ac machine by 30 electrical degrees ( $\beta =\frac{\mathrm{5\pi }}{6}={\text{150}}^0$ ). For n =1, 3, 5 calculate the fractional reduction in flux linkage due to short pitching.