A unity feedback system has the forward loop transfer function $$G\left( s \right) = {{K{{\left( {s + 2} \right)}^2}} \over {{s^2}\left( {s - 1} \right)}}$$

(a) Determine the range of $$K$$ for stable operation

(b) Determine the imaginary axis crossover points

(c) Without calculating the real axis break - away points, sketch the form of root loci for the system.

An under damped second order system having a transfer function of the form $$M\left( s \right) = {{k\omega _n^2} \over {{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}\,\,$$ has frequency response plot shown in Fig. below, then the system gain $$K$$ is __________ and the damping factor is approximately _______

The current $$i(t)$$ passing through $$10\Omega $$ resistor as shown in fig $$A$$. as a waveform as shown in fig $$B.$$ Then the reading of the d.c voltmeter is connected across $$10\Omega $$ resistor is