The positive, negative and zero sequence impedances of a three phase generator are Z₁, Z₂ and Z₀ respectively. For a line-to-line fault with fault impedance Z_f, the fault current is I_f1 = kI_f, where I_f is the fault current with zero fault impedance. The relation between Z_f and k is

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).