In the chopper circuit shown in fig the input $$dc$$ voltage has a constant value $${{V_s}}$$. the output voltage $${{V_0}}$$ is assumed ripple free. The switch $$S$$ is operated with a switching time period $$T$$ and a duty ratio $$D.$$ What is the value of $$D$$ at the boundary of continuous and discontinuous conduction of the inductor current $${i_L}$$?

A step down chopper operates from a $$dc$$ voltage source $${V_s}$$ feeds a $$dc$$ motor armature with a back $$e.m.f$$ $$\,\,{E_b}.$$ From oscilloscope traces, it is found that the current increases for time $${t_r}$$ falls to zero over time $${t_f},$$ and remains zero for time $${t_0}$$ in every chopping cycle, then the average $$dc$$ voltage across the freewheeling diode is