5.1 Problems - chapter 5

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

5.1 The full-load torque angle of a synchronous motor at rated voltage and frequency is 35 electrical degrees. Neglect the effects of armature resistance and leakage reactance. If the field current is held constant, how would the full-load torque angle be affected by the following changes in operating condition?

a. Frequency reduced 10 percent, load torque and applied voltage constant.

b. Frequency reduced 10 percent, load power and applied voltage constant.

c. Both frequency and applied voltage reduced 10 percent, load torque constant.

d. Both frequency and applied voltage reduced 10 percent, load power constant.

5.2 Design calculations show the following parameters for a three-phase, cylindrical-rotor synchronous generator:

Phase-a self-inductance $L_{\text{aa}}$ = 4.83 mH

Armature leakage inductance $L_{\mathrm{a1}}$ = 0.33 mH

Calculate the phase-phase mutual inductance and the machine synchronous inductance.

5.3 The open-circuit terminal voltage of a three-phase, 60-Hz synchronous generator is found to be 15.4 kV rms line-to-line when the field current is 420 A.

a. Calculate the stator-to-rotor mutual inductance $L_{\text{af}}$ .

b. Calculate the open-circuit terminal voltage if the field current is held constant while the generator speed is reduced so that the frequency of the generated voltage is 50 Hz.

5.4 A 460-V, 50-kW, 60-Hz, three-phase synchronous motor has a synchronous reactance of $X_S=4\text{.}\text{15}\Omega$ and an armature-to-field mutual inductance, $L_{\text{af}}$ = 83 mH. The motor is operating at rated terminal voltage and an input power of 40 kW. Calculate the magnitude and phase angle of the line-toneutral generated voltage ${\hat{E}}_{\text{af}}$ and the field current $I_f$ if the motor is operating at (a) 0.85 power factor lagging, (b) unity power factor, and (c) 0.85 power factor leading.

5.5 The motor of Problem 5.4 is supplied from a 460-V, three-phase source through a feeder whose impedance is $Z_f$ = 0.084 + j0.82 $\Omega$ . Assuming the system (as measured at the source) to be operating at an input power of 40 kW, calculate the magnitude and phase angle of the line-to-neutral generated voltage ${\hat{E}}_{\text{af}}$ and the field current $I_f$ for power factors of (a) 0.85 lagging, (b) unity, and (c) 0.85 leading.

5.6 A 50-Hz, two-pole, 750 kVA, 2300 V, three-phase synchronous machine has a synchronous reactance of 7.75 $\Omega$ and achieves rated open-circuit terminal voltage at a field current of 120 A.

a. Calculate the armature-to-field mutual inductance.

b. The machine is to be operated as a motor supplying a 600 kW load at its rated terminal voltage. Calculate the internal voltage $E_{\text{af}}$ and the corresponding field current if the motor is operating at unity power factor.

c. For a constant load power of 600 kW, write a MATLAB script to plot the terminal current as a function of field current. For your plot, let the field current vary between a minimum value corresponding to a machine loading of 750 kVA at leading power factor and a maximum value corresponding to a machine loading of 750 kVA at lagging power factor. What value of field current produces the minimum terminal current? Why?