With initial condition $$x\left( 1 \right)\,\,\, = \,\,\,\,0.5,\,\,\,$$ the solution of the differential equation, $$\,\,\,t{{dx} \over {dt}} + x = t\,\,\,$$ is

A half-controlled single-phase bridge rectifier is supplying an $$R$$-$$L$$ load. It is operated at a firing angle $$α$$ and the load current is continuous. The fraction of cycle that the freewheeling diode conducts is

The typical ratio of latching current to holding current in a $$20$$ $$A$$ thyristor is

In the circuit shown, an ideal switch $$S$$ is operated at $$100$$ $$kHz$$ with a duty ratio of $$50%$$. Given that $$\Delta {i_c}$$ is $$1.6$$ $$A$$ peak-to-peak and $${{\rm I}_0}$$ is $$5 A$$ $$dc,$$ the peak current in $$S$$ is