Question

Problem 1 (20 pts) In class, you were given the following expression for electromagnetic torque for a two-pole round- rotor synchronous machine: T = Lai'salias – 0.5ibs – 0.5ics) cos 6, +3/2libs – ick) sin 0, - (1) By assuming that is, ib. ic constitute a balance set of sinusoidal waveforms (e.gia(t) = V21 cos(wet), ibs(t) = V2I cos(wet- 27/3), and is (t) = 21 cos(wet + 2/3)]; i t is constant by definition in steady state, and r(t) = wet + 1/2 + 8 (for constant rotor speed and fixed angle 8), prove that the electromagnetic torque T.(t) is constant for a fixed 8. Hint: Use trigonometric identities of product to addition of sinusoidal functions. Problem 2 (50 pts) Simulate and study the dynamic performance of the synchronous machine we derived in class for te (0,25) s. Consider that LA = 2.3 x 10-*H, rs = 2.43 x 10-12, Lis = 4.079 x 10-H, Lifd = 54.5 x 10-SH, rfd = 0.1345 12, Lmfd = 619.7 x 10-H, Lifd = 37.8 x 10-*H, and N /N, = 16.4. These parameters pertain to a two-pole round-rotor synchronous machine which is rated 835 MVA; its rated line-to-line voltage is 26 kV. For informational purposes, the inertia constant of the machine-turbine assembly is J = 0.0658 x 100 Kg.mand the viscous damping constant is D = 7N-m-s/rad. Because we are not modeling the dynamics of the prime mover, assume that Tm= -T+T that is, the prime mover ideally balances the net torque so that angular acceleration is zero. Also assume that the excitation voltage ufa(t > 0) = 230 V, i.e., it remains constant throughout the simulation because we are not modeling the dynamics of the voltage regulator and exciter. The initial conditions of all the states will be zero other than rotor speed which will be wr(0-) = 120 rad/s. Assume also that the terminals of the machine are connected to a balanced wye-connected restive load with RL = 40.48 12 forte [0, 15)s and that RL = 2.722 forte (15,25) s. Note that this will introduce a transient to the machine around t = 15 s, e.g., a load pick up event. To earn credit, depict the three-phase voltage and current sinusoidal waveforms observed at the terminals of the machine, the field winding current, the rotor speed, and electromagnetic torque as well as electric power measured at the terminals of the machine. Discuss on the observations from the figures. Problem 3 (30 pts) Repeat problem 2 but consider the following balanced wye-connected resistive load events: RL= (40.482 te [0.15)s 2.702 te [15, 20) s 0.0022 te 20, 20+5/60) s 2.702 > 20+5/60 s Note that the terminals of the machine are in short circuit for te 20,20 +5/60) s. To earn credit, depict the three-phase voltage and current waveforms observed at the terminals of the machine, the field winding current, the rotor speed and electromagnetic torque as well as electric power measured at the terminals of the machine. Discuss on your observations from the figures, specially during the short circuit event.

Answer 1

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