A buck converter, as shown in Figure $$(a)$$ below, is working in steady state. The output voltage and the inductor current can be assumed to be ripple free. Figure $$(b)$$ shows the inductor voltage $${V_L}$$ during a complete switching interval. Assuming all devices are ideal, the duty cycle of the buck converter is ________.

A single-phase transmission line has two conductors each of 10 mm radius. These are fixed at a center-to-center distance of 1 m in a horizontal plane. This is now converted to a three-phase transmission line by introducing a third conductor of the same radius. This conductor is fixed at an equal distance D from the two single-phase conductors. The three-phase line is fully transposed. The positive sequence inductance per phase of the three phase system is to be 5% more than that of the inductance per conductor of the single phase system. The distance D, in meters, is ________.

A three-phase cable is supplying $$800$$ kW and $$600$$ kVAr to an inductive load. It is intended to supply an additional resistive load of $$100$$ kW through the same cable without increasing the heat dissipation in the cable, by providing a three-phase bank of capacitors connected in star across the load. Given the line voltage is $$3.3$$ kV, $$50$$ Hz, the capacitance per phase of the bank expressed in microfarads, is ________.

A 30 MVA, 3-phase, 50Hz, 13.8 kV, star-connected synchronous generator has positive, negative and zero sequence reactances, 15%, 15% and 5% respectively. A reactance (X_n) is connected between the neutral of the generator and ground. A double line to ground fault takes place involving phases ‘b’ and ‘c’, with a fault impedance of j0.1 p.u. The value of X_n (in p.u.) that will limit the positive sequence generator current to 4270 A is __________.