Consider an overhead transmission line with $$3$$-phase, $$50$$ $$Hz$$ balanced system with conductors located at the vertices of an equilateral triangle of length $$\,{D_{ab}} = {D_{bc}} = {D_{ca}} = 1\,\,\,$$ m as shown in figure below. The resistance of the conductors are neglected. The geometric mean radius (GMR) of each conductor is $$0.01$$ $$m$$. Neglect the effect of ground, the magnitude of positive sequence reactance in $$\,\Omega $$/$$km$$ (rounded off to three decimal places) is ________________.

A load is supplied by a $$230$$ $$V,$$ $$50$$ $$Hz$$ source. The active power $$P$$ and the reactive power $$Q$$ consumed by the load are such that $$1$$ $$kW$$ $$ \le P \le 2\,kW\,\,$$ and $$\,1\,\,kVAR \le 2\,\,kVAR.\,\,$$ A capacitor connected across the load for power factor correction generates $$1$$ $$kVAR$$ reactive power. The worst case power factor after power factor correction 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 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 ________.