A 50 Hz, 275 kV line of length 400 km has the following parameters:

Resistance, R = 0.035 $$\Omega$$/km;

Inductance, L = 1 mH/km;

Capacitance, C = 0.01 $$\mu$$F/km;

The line is represented by the nominal-$$\pi$$ model. With the magnitudes of the sending end and the receiving end voltages of the line (denoted by $$V_S$$ and $$V_R$$, respectively) maintained at 275 kV, the phase angle difference ($$\theta$$) between $$V_S$$ and $$V_R$$ required for maximum possible active power to be delivered to the receiving end, in degree is ___________ (Round off to 2 decimal places).

The three-bus power system shown in the figure has one alternator connected to bus 2 which supplies 200 MW and 40 MVAr power. Bus 3 is infinite bus having a voltage of magnitude $$|{V_3}| = 1.0$$ p.u. and angle of $$-15^\circ$$. A variable current source, $$|I|\angle \phi $$ is connected at bus 1 and controlled such that the magnitude of the bus 1 voltage is maintained at 1.05 p.u. and the phase angle of the source current, $$\phi = {\theta _1} \pm {\pi \over 2}$$, where $$\theta_1$$ is the phase angle of the bus 1 voltage. The three buses can be categorized for load flow analysis as

The two-bus power system shown in figure (i) has one alternator supplying a synchronous motor load through a Y-$$\Delta$$ transformer. The positive, negative and zero-sequence diagrams of the system are shown in figures (ii), (iii) and (iv), respectively. All reactances in the sequence diagrams are in p.u. For a bolted line-to-line fault (fault impedance = zero) between phases 'b' and 'c' at bus 1, neglecting all pre-fault currents, the magnitude of the fault current (from phase 'b' to 'c') in p.u. is ____________ (Round off to 2 decimal places).

A continuous-time system that is initially at rest is described by

$${{dy(t)} \over {dt}} + 3y(t) = 2x(t)$$,

where $$x(t)$$ is the input voltage and $$y(t)$$ is the output voltage. The impulse response of the system is