In the figure, the voltages are

$${v_1}\left( t \right) = 100\cos \left( {\omega t} \right)$$

$${v_2}\left( t \right) = 100\cos \left( {\omega t + {\pi \over {18}}} \right)$$

and $${v_3}\left( t \right) = 100\cos \left( {\omega t + {\pi \over {36}}} \right)$$.

The circuit is in sinusoidal steady state, and R << $${\omega L}$$. P₁, P₂ and P₃ are the average power outputs. Which one of the following statements is true?

Consider the two bus power system network with given loads as shown in the figure. All the values shown in the figure are in per unit. The reactive power supplied by generator G₁ and G₂ are Q_G1 and Q_G2 respectively. The per unit values of Q_G1, Q_G2, and line reactive power loss (Q_loss) respectively are

The series impedance matrix of a short three-phase transmission line in phase coordinates is $$\left[ {\matrix{ {{Z_s}} & {{Z_m}} & {{Z_m}} \cr {{Z_m}} & {{Z_s}} & {{Z_m}} \cr {{Z_m}} & {{Z_m}} & {{Z_s}} \cr } } \right]$$.

If the positive sequence impedance is (1 + 𝑗 10) $$\Omega $$, and the zero sequence is (4 + 𝑗 31) $$\Omega $$, then the imaginary part of Z_m (in $$\Omega $$) is ______(up to 2 decimal places).

The positive, negative and zero sequence impedances of a 125 MVA, three-phase, 15.5 kV, star-grounded, 50 Hz generator are 𝑗0.1 pu, j0.05 pu and j0.01 pu respectively on the machine rating base. The machine is unloaded and working at the rated terminal voltage. If the grounding impedance of the generator is j0.01 pu, then the magnitude of fault current for a b-phase to ground fault (in kA) is __________ (up to 2 decimal places).