8.1 Problems - chapter 8

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

8.1 Repeat Example 8.1 for a machine identical to that considered in the example except that the stator pole-face angle is, β = 45°.

8.2 In the paragraph preceeding Eq. 8.1, the text states that "under the assumption of negligible iron reluctance the mutual inductances between the phases of the doubly-salient VRM of Fig. 8.lb will be zero, with the exception of a small, essentially constant component gyyassociated with leakage flux." Neglect any leakage flux effects and use magnetic circuit techniques to show that this statement is true.

8.3 Use magnetic-circuit techniques to show that the phase-to-phase mutual inductance in the 6/4 VRM of Fig. 8.5 is zero under the assumption of infinite rotor- and stator-iron permeability. Neglect any contributions of leakage flux.

8.4 A 6/4 VRM of the form of Fig. 8.5 has the following properties:

Stator pole angle $\beta ={\text{30}}^o$

Rotor pole angle $\alpha ={\text{30}}^o$

Air-gap length g = 0.35 mm

Rotor outer radius R = 5.1 cm

Active length D = 7 cm

This machine is connected as a three-phase motor with opposite poles connected in series to form each phase winding. There are 40 turns per pole (80 turns per phase). The rotor and stator iron can be considered to be of infinite permeability and hence mutual-inductance effects can be neglected.

a. Defining the zero of rotor angle ${\theta }_m$ at the position when the phase-1 inductance is maximum, plot and label the inductance of phase 1 as a function of rotor angle.

b. On the plot of part (a), plot the inductances of phases 2 and 3.

c. Find the phase-1 current $I_0$ which results in a magnetic flux density of 1.0 T in the air gap under the phase-1 pole face when the rotor is in a position of maximum phase-1 inductance.

d. Assuming that the phase-1 current is held constant at the value found in part (c) and that there is no current in phases 2 and 3, plot the torque as a function of rotor position. The motor is to be driven from a three-phase current-source inverter which can be switched on or off to supply either zero current or a constant current of magnitude $I_0$ in phases 2 and 3; plot the torque as a function of rotor position.

e. Under the idealized assumption that the currents can be instantaneously switched, determine the sequence of phase currents (as a function of rotor position) that will result in constant positive motor torque, independent of rotor position.

f. If the frequency of the stator excitation is such that a time $T_0$ = 35 msec is required to sequence through all three phases under the excitation conditions of part (e), find the rotor angular velocity and its direction of rotation.

8.5 In Section 8.2, when discussing Fig. 8.5, the text states: "In addition to the fact that there are not positions of simultaneous alignment for the 6/4 VRM, it can be seen that there also are no rotor positions at which only a torque of a single sign (either positive or negative) can be produced." Show that this statement is true.