A composite conductor consists of three conductors of radius $$R$$ each. The conductors are arranged as shown below. The geometric mean radius (GMR) (in cm) of the composite conductor is kR. The value of k is ______.

A two bus power system shown in figure supplies load of $$1.0+j0.5$$ $$p.u$$

The value of $${V_1}$$ in $$p.u$$ and $${\delta _2}$$ respectively are

A distribution feeder of $$1$$ $$km$$ length having resistance, but negligible reactance, is fed from both the ends by $$400$$ $$V,$$ $$50$$ $$Hz$$ balanced sources. Both voltage sources $${S_1}$$ and $${S_2}$$ are in phase. The feeder supplies concentrated loads of unity power factor as shown in the figure.

The contributions of $${S_1}$$ and $${S_2}$$ in $$100$$ $$A$$ current supplied at location $$P$$ respectively, are

For a $$400$$ $$km$$ long transmission line, the series impedance is $$(0.0 + j0.5)$$ $$\Omega /km$$ and the shunt admittance is $$(0.0 + j5.0)$$ $$\mu \,mho/km.$$ The magnitude of the series impedance (in $$\Omega $$) of the equivalent $$\pi $$ circuit of the transmission line is __________.