A differential equation $$\,\,{{di} \over {dt}} - 0.21 = 0\,\,$$ is applicable over $$\,\, - 10 < t < 10.\,\,$$ If $$i(4)=10,$$ then $$i(-5)$$ is

In the given rectifier, the delay angle of the thyristor $${T_1}$$ measured from the positive going zero crossing of $${V_s}$$ is $${30^ \circ }$$. If the input voltage $${V_s}$$ is $$100$$ $$sin$$ $$\left( {100\,\pi t} \right)$$ $$V,$$ the average voltage across $$R$$ (in Volt) under steady-state is ________.

For the switching converter shown in the following figure, assume steady-state operation. Also assume that the components are ideal, the inductor current is always positive and continuous and switching period is $${T_s}.$$ If the voltage $${V_L}$$ is as shown, the duty cycle of the switch $$S$$ is __________.

A $$3$$-bus power system network consists of $$3$$ transmission lines. The bus admittance matrix of the uncompensated system is
$$\left[ {\matrix{ { - j6} & {j3} & {j4} \cr {j3} & { - j7} & {j5} \cr {j4} & {j5} & { - j8} \cr } } \right]\,pu$$
If the shunt capacitance of all transmission lines is $$50$$% compensated, the imaginary part of the $$3$$^rd row $$3$$^rd column element (in $$pu$$) of the bus admittance matrix after compensation is